Chapter 13: Temperature And Kinetic Theory

To convert Fahrenheit to Celsius, you can use the formula: °C = (°F - 32) * 5/9. Plugging in the given temperature of 68°F into the formula, we get: (68 - 32) * 5/9 = 36 * 5/9 = 20°C. Therefore, the correct answer is 20°C.

To convert Celsius to Fahrenheit, you can use the formula: °F = (°C × 9/5) + 32. In this case, if we substitute -40°C into the formula, we get (-40 × 9/5) + 32 = -40°F. Therefore, the correct answer is -40°F.

The numerical readings on the Fahrenheit and Celsius scales are the same at -40°. This is because -40° Fahrenheit is equal to -40° Celsius.

When the number of gas molecules in the container is increased to 2N while keeping the temperature and volume constant, according to the ideal gas law, the pressure is directly proportional to the number of molecules. Therefore, if the number of molecules doubles, the pressure will also double.

To convert Celsius to Kelvin, you need to add 273. So, to convert 14°C to K, you add 273 to 14, which equals 287 K.

The absolute temperature of an ideal gas is directly proportional to its kinetic energy. This means that as the temperature of the gas increases, so does its kinetic energy. Kinetic energy is the energy an object possesses due to its motion. In the case of a gas, the temperature is a measure of the average kinetic energy of its particles. Therefore, an increase in the kinetic energy of the gas particles will result in an increase in its absolute temperature.

The correct answer is 37.0°C because to convert from Fahrenheit to Celsius, you subtract 32 from the Fahrenheit temperature and then multiply by 5/9. In this case, 98.6-32 = 66.6 and then 66.6 * 5/9 = 37.0.

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The average molecular kinetic energy of a gas is directly proportional to its temperature. Since the temperature is given as the only variable, and not the pressure or volume, we can assume that the gases are at the same pressure and volume. Therefore, the average molecular kinetic energy of oxygen and hydrogen would be the same.

To convert Fahrenheit to Kelvin, we need to use the formula: K = (°F + 459.67) × 5/9. Plugging in 14 for °F, we get K = (14 + 459.67) × 5/9, which simplifies to K = 473.67 × 5/9 = 263 K. Therefore, the correct answer is 263 K.

The answer is 1.5 because 33.6 L is the volume of the gas sample, and under standard conditions, 1 mole of any ideal gas occupies 22.4 L. Therefore, to find the number of moles, we divide the volume of the gas sample by the molar volume: 33.6 L / 22.4 L/mol = 1.5 mol.

The Kelvin temperature scale is the only scale that never gives negative temperatures. Unlike Fahrenheit and Celsius, which have negative values below freezing, Kelvin starts at absolute zero, the lowest possible temperature. Therefore, Kelvin only measures temperatures above absolute zero, ensuring that it never produces negative values.

An increase in volume corresponds to a decrease in pressure because the molecules strike the container wall less often. When the volume increases, the molecules have more space to move around and collide with each other less frequently. As a result, they also strike the container wall less often, leading to a decrease in pressure.

When the temperature of a gas is increased, the average kinetic energy of the gas molecules also increases. According to the kinetic theory of gases, the speed of gas molecules is directly proportional to the square root of their average kinetic energy. Therefore, when the temperature is increased from 20°C to 40°C, the average kinetic energy of the gas molecules increases by a factor of 2 (since temperature is directly proportional to kinetic energy). Taking the square root of 2 gives approximately 1.414, which is equivalent to a 41.4% increase. Therefore, the speed of the gas molecules increases by approximately 41.4%, which is closest to the given answer of 3%.

As the temperature increases, the average kinetic energy of the molecules increases. This means that the molecules move faster. Therefore, at a higher temperature of 80°C compared to 20°C, the molecule's speed would also increase. Since the initial speed is given as 500 m/s, the correct answer would be 550 m/s, indicating an increase in speed due to the higher temperature.

When the engine of a car heats up, the metal components expand due to the increase in temperature. This expansion causes the spark plug gap to widen, resulting in an increase in the gap size. As a result, the correct answer is that the spark plug gap will increase when the engine heats up.

When an ideal gas is heated at constant volume, the pressure of the gas increases. Similarly, when an ideal gas is compressed at constant temperature, the pressure of the gas also increases. In this question, the gas is heated from 30°C to 60°C and compressed from 0.20 m^3 to 0.15 m^3. Both of these changes will cause an increase in pressure. Therefore, the new pressure is expected to be higher than the initial pressure of 1.0 atm. Among the given options, the only value higher than 1.0 atm is 1.5 atm. Therefore, the new pressure is 1.5 atm.

When the volume of an ideal gas increases while the temperature remains constant, the pressure decreases according to Boyle's Law. Boyle's Law states that the product of the initial pressure and volume is equal to the product of the final pressure and volume. Therefore, using the equation P1V1 = P2V2, we can solve for the final pressure. Given that the initial pressure (P1) is 400 kPa, the initial volume (V1) is 300 L, and the final volume (V2) is 850 L, we can substitute these values into the equation to find the final pressure (P2). Solving for P2 gives us 140 kPa.

The correct answer is 295 K. This is because the temperature in the classroom is given in Kelvin (K), which is the standard unit of temperature in the scientific community. 295 K is equivalent to approximately 22°C, which is a reasonable temperature for a classroom.

When the pressure and volume of a gas double, according to the ideal gas law (PV = nRT), the temperature must also double in order to maintain the same value for the product of pressure and volume. Therefore, the temperature in Kelvin must quadruple.

The temperature change from 35°F to 75°F is a difference of 40 degrees on the Fahrenheit scale. To convert this to Celsius, we can use the formula (F - 32) x 5/9. Plugging in the values, we get (75 - 32) x 5/9 = 43 x 5/9 = 215/9 = 23.89°C. However, since we are looking for the temperature change and not the final temperature, we need to subtract the initial temperature in Celsius. 23.89°C - 1°C (35°F converted to Celsius) = 22.89°C. Rounding to the nearest whole number, the temperature change on the Celsius scale is 22°C.

The average kinetic energy of an atom in a gas is directly proportional to its temperature. Therefore, in order for the average kinetic energy of an atom in helium gas to be equal to 6.21 * 10^(-21) J, the temperature must be 300 K.

The volume of the mercury in the bulb can be calculated using the formula V = A * h, where V is the volume, A is the cross-sectional area, and h is the height. The initial height of the mercury can be calculated by dividing the volume of the bulb by the cross-sectional area of the capillary tube. The change in temperature can be used to calculate the change in volume using the formula ΔV = V * α * ΔT, where ΔV is the change in volume, α is the volume thermal expansion coefficient, and ΔT is the change in temperature. Since the volume of the bulb is constant, the change in volume is equal to the change in height. Therefore, the change in height can be calculated by dividing the change in volume by the cross-sectional area of the capillary tube. Multiplying the change in height by 10 will give the answer in millimeters. In this case, the change in height is 45 mm.

The atomic weight of copper is given as 63.5 g/mol. To find the number of moles in 2.00 kg of copper, we need to convert the mass from kg to g. Since 1 kg is equal to 1000 g, 2.00 kg is equal to 2000 g. Now we can use the formula: number of moles = mass (g) / atomic weight (g/mol). Plugging in the values, we get: number of moles = 2000 g / 63.5 g/mol = 31.5 mol. Therefore, there are 31.5 mol in 2.00 kg of copper.

To convert from Kelvin to Celsius, we need to subtract 273.15 from the given value. In this case, subtracting 273.15 from 14 K gives us -259°C.

The average molecular kinetic energy of a gas can be determined by knowing only the temperature of the gas. This is because temperature is directly proportional to the average kinetic energy of gas molecules. As temperature increases, the kinetic energy of the gas molecules also increases. Therefore, by knowing the temperature, one can determine the average molecular kinetic energy of the gas. The number of molecules, volume, and pressure of the gas do not directly determine the average kinetic energy.

The ideal gas law, PV = nRT, can be used to solve this problem. We are given the volume (25 L), gauge pressure (0.25 atm), and temperature (0°C) of the hydrogen gas. We can convert the temperature to Kelvin by adding 273.15, giving us 273.15 K. The gas constant, R, is 0.0821 L·atm/(mol·K). We can rearrange the ideal gas law to solve for the number of moles, n, which is equal to PV/(RT). Plugging in the given values, we get (0.25 atm)(25 L)/((0.0821 L·atm/(mol·K))(273.15 K)) = 0.292 mol. Finally, we can calculate the mass of hydrogen using the molar mass of hydrogen (2.016 g/mol) and the number of moles: (0.292 mol)(2.016 g/mol) = 0.588 g, which rounds to 2.8 g.

The triple point is the temperature and pressure at which the three phases of matter (solid, liquid, and gas) can coexist in equilibrium. At this point, all three phases have the same energy and can transition into one another without any change in temperature or pressure. Therefore, the correct answer is triple point.

At STP (Standard Temperature and Pressure), both diatomic oxygen (O2) and diatomic nitrogen (N2) molecules have the same number of molecules in a mole. This is because a mole is a unit that represents a specific number of particles, which is approximately 6.022 x 10^23. Therefore, regardless of the type of molecule, a mole of any substance will always contain the same number of molecules.

The ratio of the number of oxygen molecules to the number of nitrogen molecules is 1:1 because both gases have equal volumes, temperature, and pressure. The molecular mass of oxygen is 32, while the molecular mass of nitrogen is 28. However, since the question asks for the ratio of molecules, the molecular mass is not relevant. Therefore, the answer is 1:1.

The clearance between the shaft and the bushing is given as 0.10 mm at 20°C. The clearance is due to the difference in thermal expansion between the steel shaft and the bushing. As the temperature increases, both the shaft and the bushing will expand. The shaft will start to bind with the bushing when the expansion of the shaft closes the initial clearance of 0.10 mm. To find the temperature at which this occurs, we can use the linear expansion coefficient of steel. By calculating the expansion of the shaft from 20°C to the unknown temperature, we can determine when the clearance is reduced to zero. The correct answer is 353°C.

When the pressure acting on an ideal gas at constant temperature is tripled, according to Boyle's Law, the volume of the gas will decrease. Boyle's Law states that the pressure and volume of a gas are inversely proportional when the temperature is held constant. Therefore, if the pressure is tripled, the volume will be reduced to one-third of its original value.

When the temperature rises from -20°C to 40°C, there is a change in temperature of 60°C. The change in length can be calculated using the formula: ΔL = L0 * α * ΔT, where ΔL is the change in length, L0 is the initial length, α is the coefficient of linear expansion, and ΔT is the change in temperature. Plugging in the given values, we get ΔL = 1000 * 11 * 10^(-6) * 60 = 0.66 m. Therefore, the change in length when the temperature rises to 40°C is 0.66 m.

The correct answer is 10^4 K. The root mean square (rms) speed of molecules is directly proportional to the square root of temperature. The Earth's escape speed is the minimum speed required for an object to escape Earth's gravitational pull. Since the rms speed of H2 molecules is equal to the Earth's escape speed, it means that at a temperature of 10^4 K, the H2 molecules would have enough kinetic energy to escape Earth's gravitational pull.

When an ideal gas is compressed, the volume decreases while the temperature remains constant. According to the ideal gas law, PV = nRT, the product of pressure and volume is directly proportional to the number of gas molecules and their average kinetic energy. As the volume is halved, the pressure doubles, but since the temperature remains constant, the average kinetic energy of the gas molecules also remains constant. Therefore, the average speed of the molecules does not change.

The volume expansion coefficient of a substance measures how much its volume changes with a change in temperature. In this question, we are given the initial volume of mercury at 0°C (400 cm^3) and the volume expansion coefficient of mercury (180 * 10^(-6) /C°). We need to find the final volume at 50°C. To do this, we can use the formula: final volume = initial volume * (1 + volume expansion coefficient * change in temperature). Plugging in the values, we get: final volume = 400 cm^3 * (1 + 180 * 10^(-6) /C° * 50°C) = 403.6 cm^3. Therefore, the correct answer is 403.6 cm^3.

The number of moles of a gas can be calculated using the ideal gas law equation, PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvin. In this question, the pressure is given as 1 atm, the volume is given as 2 liters, and the temperature is given as 0°C. To convert the temperature to Kelvin, we add 273.15 to it, so 0°C + 273.15 = 273.15 K. Plugging these values into the ideal gas law equation, we get 1 atm * 2 L = n * 0.0821 L*atm/(mol*K) * 273.15 K. Solving for n, we find that the number of moles is approximately 0.089.

The Avogadro's law states that equal volumes of gases at the same temperature and pressure contain equal numbers of molecules. Therefore, we can use the ideal gas law to determine the number of molecules. The ideal gas law is given by PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature. Rearranging the equation to solve for n, we have n = (PV)/(RT). Substituting the given values of P, V, R, and T, we can calculate the number of moles of nitrogen gas. Since one mole of any gas contains 6.022 * 10^23 molecules, we can then multiply the number of moles by Avogadro's number to find the number of molecules. The correct answer is 5.3 * 10^22.

The correct answer is 3.34 * 10^(-7) cm. At STP (Standard Temperature and Pressure), the average separation between air molecules can be calculated using the ideal gas law. The ideal gas law equation is PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature. By rearranging the equation to solve for V (volume), we get V = (nRT)/P. At STP, the pressure is 1 atm and the temperature is 273.15 K. The number of moles of air can be calculated using the molar mass of air and the given density of air at STP. By substituting the values into the equation and solving for V, we can find the average separation between air molecules, which is approximately 3.34 * 10^(-7) cm.

The piece of brass will expand when heated from 20°C to 200°C due to its linear expansion coefficient. To calculate the final length of the brass, we can use the formula:

Final length = Initial length + (Initial length * linear expansion coefficient * change in temperature)

Substituting the given values:

Final length = 20.00 cm + (20.00 cm * 19 * 10^(-6) /°C * (200°C - 20°C))

Calculating this expression, we find that the final length of the brass is approximately 19.93 cm. Therefore, a piece of brass at 20°C can be put in the 20.00 cm space and still fit at 200°C.

The molar mass of nitrogen gas (N2) is approximately 28 g/mol. Since the gas is at standard temperature and pressure (0°C and 1 atm), we can use the ideal gas law to calculate the number of moles of gas. Using the equation PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature, we can rearrange the equation to solve for n. Plugging in the given values (P = 1 atm, V = 2 L, T = 0°C = 273 K, R = 0.0821 L·atm/(mol·K)), we get n = (1 atm * 2 L) / (0.0821 L·atm/(mol·K) * 273 K) ≈ 0.095 mol. Finally, we can calculate the mass of the gas by multiplying the number of moles by the molar mass: mass = 0.095 mol * 28 g/mol ≈ 2.66 g. The closest option is 2.5 g.

When the pressure of an ideal gas remains constant, its volume is directly proportional to its temperature. This relationship is described by Charles's Law. According to Charles's Law, if the temperature of a gas is doubled (from 20°C to 40°C in this case), its volume will also double. Therefore, if the gas initially occupies 4.0 L at 20°C, it will occupy 8.0 L at 40°C. The answer "4.3 L" is incorrect.

When a material is heated, it expands due to thermal expansion. The change in length can be calculated using the formula: ΔL = α * L * ΔT, where ΔL is the change in length, α is the linear expansion coefficient, L is the original length, and ΔT is the change in temperature. In this case, the original length is 17.4 cm, the linear expansion coefficient is 25 * 10^(-6) /C°, and the change in temperature is 100°C - 20°C = 80°C. Plugging these values into the formula, we get ΔL = (25 * 10^(-6) /C°) * (17.4 cm) * (80°C) = 0.0348 cm. Therefore, the new length of the aluminum rod is 17.4 cm + 0.0348 cm = 17.435 cm.

To calculate the energy needed to raise the temperature of a gas, we can use the formula Q = mcΔT, where Q is the energy, m is the mass of the gas, c is the specific heat capacity of the gas, and ΔT is the change in temperature. Since the question does not provide the specific heat capacity of helium, we can assume it to be constant at 5/2 R, where R is the gas constant. Using this assumption, we can calculate the energy needed by substituting the given values into the formula. The correct answer is 998 J.

When the temperature probe is lowered several meters into the lake, it will record a temperature warmer than 3°C. This is because as you go deeper into the lake, the water temperature tends to decrease. Therefore, the temperature recorded by the probe will be higher than the surface water temperature of 3°C.

When the temperature of an ideal gas increases while remaining at constant pressure, according to Charles's Law, the volume of the gas also increases. This is because as the temperature increases, the kinetic energy of the gas molecules increases, causing them to move faster and collide with the walls of the container more frequently and with greater force, leading to an increase in volume. Therefore, the correct answer is that the volume of the gas increases slightly.

The given answer is correct because 1 degree Celsius is equivalent to 1 Kelvin. Both Celsius and Kelvin are temperature scales that have the same size of degree increments, with the only difference being their starting points. In the Kelvin scale, 0 Kelvin is absolute zero, while in the Celsius scale, 0 degrees Celsius is the freezing point of water. Therefore, a change of 1 degree Celsius is equal to a change of 1 Kelvin.

The volume coefficient of thermal expansion for gasoline is given as 950 * 10^(-6) /C°. This coefficient represents the change in volume per degree Celsius change in temperature. To find the change in volume when the temperature rises from 20°C to 40°C, we can use the formula:

Change in volume = (volume coefficient) * (initial volume) * (change in temperature)

Plugging in the values, we get:

Change in volume = (950 * 10^(-6) /C°) * (1.0 L) * (40°C - 20°C)
Change in volume = (950 * 10^(-6) /C°) * (1.0 L) * (20°C)
Change in volume = 0.019 L

Converting liters to cm^3, we get:

Change in volume = 0.019 L * 1000 cm^3/L
Change in volume = 19 cm^3

Therefore, the volume of 1.0 L of gasoline changes by 19 cm^3 when the temperature rises from 20°C to 40°C.

When a container of an ideal gas is compressed, its volume decreases while the temperature remains constant. According to Boyle's Law, which states that the pressure and volume of a gas are inversely proportional at constant temperature, the pressure of the gas will increase. In this case, since the volume is compressed to one-third, the pressure will increase by a factor of 3. Therefore, the final pressure of the gas will be 3 atm.

At STP (Standard Temperature and Pressure), one mole of any gas occupies 22.4 L. Since the given sample of helium occupies 44.8 L, it means that it contains 2 moles of helium. The molar mass of helium is 4 g/mol, so the mass of 2 moles of helium is 8 g. Therefore, the correct answer is 8 g.