Thermodynamics Temperature and Thermal Expansion

To find the change in length (ΔL) of the steel bridge due to temperature change, we use the formula ΔL = L₀ * α * ΔT. Here, L₀ is the original length (30 m), α is the coefficient of linear expansion (12 × 10⁻⁶/°C), and ΔT is the change in temperature (55°C). Plugging in the values: ΔL = 30 m * 12 × 10⁻⁶/°C * 55°C = 0.0198 m, which converts to 19.8 mm. This calculation shows how materials expand with temperature, leading to the observed change in length.

To find the change in internal energy (ΔU₂), we use the first law of thermodynamics, which states ΔU = Q + W. For Process 2, we have Q₂ = -250 J (heat lost) and W₂ = -100 J (work done on the system). Plugging in these values: ΔU₂ = Q₂ + W₂ = -250 J - 100 J = -350 J. However, since work is done on the system, it should be considered as positive in terms of energy gained. Therefore, ΔU₂ = -250 J + 100 J = -150 J, which accounts for the energy changes accurately.

To find the change in internal energy (ΔU₁) for Process 1, we use the first law of thermodynamics, which states that ΔU = Q + W. Here, Q₁ (heat added) is +800 J and W₁ (work done on the system) is +300 J. Adding these values together gives ΔU₁ = 800 J + 300 J = 1100 J. However, the work done by the system is usually considered negative, so we take it as -300 J. Thus, ΔU₁ = 800 J - 300 J = +500 J.

In the First Law of Thermodynamics, when a system does work on its surroundings, it means energy is being transferred from the system to the external environment. This transfer of energy is represented by a positive value for W, indicating that the system is expending energy to perform work, such as moving a piston or lifting an object. Therefore, the surroundings are the entities that receive this energy, making them the focus of the work done by the system.

To find the change in internal energy (ΔU), we use the first law of thermodynamics, which states that ΔU = Q - W, where Q is the heat absorbed and W is the work done by the gas. Here, the gas absorbs 600 J of heat (Q = +600 J) and does 250 J of work (W = +250 J). Substituting these values, we get ΔU = 600 J - 250 J = +350 J. Thus, the change in internal energy is +350 J, indicating that the internal energy of the gas increased.

In the First Law of Thermodynamics, Q represents the heat transfer into or out of a system. When heat enters the system, it increases the internal energy, which is represented by a positive value for Q. This aligns with the principle that energy cannot be created or destroyed but can change forms. Therefore, when heat is absorbed by the system, it contributes to the overall energy balance, resulting in a positive Q value.

The First Law of Thermodynamics states that energy cannot be created or destroyed, only transformed. In this equation, ΔU represents the change in internal energy of a system, Q is the heat added to the system, and W is the work done by the system. The equation captures the principle of conservation of energy, indicating that the change in internal energy is equal to the heat energy input minus the work output. Thus, it accurately reflects the relationship between heat, work, and energy in thermodynamic processes.

The Zeroth Law of Thermodynamics establishes a fundamental principle regarding thermal equilibrium. It states that if two systems (A and B) are each in thermal equilibrium with a third system (C), then A and B must also be in thermal equilibrium with each other. This implies that they share the same temperature, as thermal equilibrium is defined by the absence of heat transfer between systems. Therefore, if A and C, and B and C are at the same temperature, it follows that A and B must also have the same temperature.

To find the change in length (ΔL) of the brass wire due to thermal expansion, we use the formula ΔL = L₀ * α * ΔT. Here, L₀ is the original length (2.5 m), α is the coefficient of linear expansion (19 × 10⁻⁶/°C), and ΔT is the change in temperature (80°C). Substituting the values, we calculate ΔL = 2.5 m * 19 × 10⁻⁶/°C * 80°C, which results in ΔL = 0.0038 m or 3.8 mm. This shows how temperature increases lead to the expansion of materials.

To find the change in volume (ΔV) of the aluminum block due to temperature change, we use the formula ΔV = V₀ * β * ΔT. Here, V₀ is the initial volume (0.80 m³), β is the coefficient of volumetric expansion (69 × 10⁻⁶/°C), and ΔT is the change in temperature (50°C). Plugging in the values, we calculate ΔV as follows: ΔV = 0.80 m³ * (69 × 10⁻⁶/°C) * 50°C, which results in ΔV = 2.76 × 10⁻³ m³. This indicates how much the volume expands with the temperature increase.

To calculate the change in length (ΔL) of the steel rail due to thermal expansion, we use the formula ΔL = L₀ * α * ΔT. Here, L₀ is the original length (12.0 m), α is the coefficient of linear expansion (12 × 10⁻⁶/°C), and ΔT is the change in temperature (40°C). Substituting these values, we find ΔL = 12.0 m * (12 × 10⁻⁶/°C) * 40°C, which simplifies to 5.76 mm. This result indicates how much the steel rail expands when subjected to a temperature increase.

In isotropic solids, the coefficient of volume expansion (β) relates to the linear coefficient of thermal expansion (α). When a solid is heated, it expands in three dimensions, leading to a volumetric change. The relationship between these coefficients is given by β = 3α because the volume expansion is effectively the sum of the expansions in each of the three orthogonal directions. Thus, for isotropic materials, the volume expansion is approximately three times the linear expansion coefficient.

The formula ΔL = αL₀ΔT describes how a solid material expands when its temperature changes. Here, ΔL represents the change in length, α is the coefficient of linear expansion specific to the material, L₀ is the original length, and ΔT is the change in temperature. This equation is fundamental in physics and engineering, as it quantifies the relationship between temperature variations and physical dimensions, ensuring accurate predictions of material behavior under thermal stress.

Heat is a form of energy transfer that occurs when there is a difference in temperature between two systems or objects. When one area is warmer than another, energy moves from the hotter region to the cooler one until thermal equilibrium is reached. This process is fundamental in thermodynamics, illustrating how energy flows and affects matter. The concept of heat emphasizes the importance of temperature gradients in driving energy transfer, making "temperature difference" the key factor in defining heat.

Metal feels colder than wood at the same temperature because it is a better conductor of heat. When you touch metal, it rapidly draws heat away from your skin, creating a sensation of coldness. In contrast, wood is a poorer conductor, meaning it retains heat better and does not absorb as much warmth from your hand. This difference in thermal conductivity explains why metal feels colder, even though both materials are at the same temperature.

To convert a temperature rise from Celsius to Fahrenheit, use the formula: \( F = C \times \frac{9}{5} \). For a 20°C increase, this calculation is \( 20 \times \frac{9}{5} = 36°F \). Thus, a rise of 20°C corresponds to a rise of 36°F, reflecting the difference in the scales where 1°C equals 1.8°F.

To convert Celsius to Kelvin, you need to account for the difference in their starting points. The Kelvin scale begins at absolute zero, which is equivalent to -273.15 degrees Celsius. Therefore, to convert a temperature from Celsius to Kelvin, you simply add 273.15 to the Celsius value. This adjustment ensures that the temperature is expressed in the Kelvin scale, which is essential for scientific calculations and measurements.

To convert Fahrenheit to Celsius, you can use the formula: \(C = \frac{(F - 32) \times 5}{9}\). Plugging in 95°F gives \(C = \frac{(95 - 32) \times 5}{9} = \frac{63 \times 5}{9} = \frac{315}{9} \approx 35\). Thus, 95°F is approximately 35°C, making this the correct conversion.

To convert a temperature from Celsius to Fahrenheit, the formula used is F = (9/5)C + 32. This formula works by first multiplying the Celsius temperature by 9/5, which scales the temperature appropriately, and then adding 32 to adjust for the difference in the zero points of the two scales. This conversion accounts for the fact that the freezing point of water is 0°C and 32°F, while the boiling point is 100°C and 212°F, maintaining the correct relationship between the two temperature scales.

Equal temperature between two objects indicates that they are in thermal equilibrium, meaning there is no net heat transfer between them. This state signifies that the objects have reached a balance in their thermal energy, leading to the same thermal state. However, it does not imply equal mass, internal energy, or material composition, as these factors can vary independently of temperature. Thus, the concept of equal temperature is specifically tied to the thermal state of the objects involved.