Problem 28 - University Physics with Modern Physics

Question

Three moles of an ideal monatomic gas expands at a con-stant pressure of 2.50 atm; the volume of the gas changes from \(3.20 \times 10^{-2} \mathrm{m}^{3}\) to \(4.50 \times 10^{-2} \mathrm{m}^{3} .\) (a) Calculate the initial and final temperatures of the gas. (b) Calculate the amount of work the gas does in expanding. (c) Calculate the amount of heat added to the gas. (d) Calculate the change in internal encrgy of the gas.

Step-by-Step Solution

Answer

(a) Initial: ~325.3 K, Final: ~457.1 K; (b) Work: 3296 J; (c) Heat: 214720 J; (d) Energy Change: 154720 J.

Step 1: Convert Pressure to SI Units

First, convert the pressure from atmospheres to Pascals for use in calculations. Given that \(1 \text{ atm} = 101325 \text{ Pa}\), we multiply: \(2.50 \text{ atm} \times 101325 \text{ Pa/atm} = 253312.5 \text{ Pa} \approx 2.53 \times 10^5 \text{ Pa}\).

Step 2: Calculate Initial and Final Temperature

Use the ideal gas law \(PV = nRT\) to find the initial and final temperatures. **Initial Temperature:**\[P V_i = n R T_i\]\[T_i = \frac{P V_i}{n R} = \frac{(2.53 \times 10^5 \text{ Pa})(3.20 \times 10^{-2} \text{ m}^3)}{3 \times 8.314 \text{ J/mol K}} \approx 325.3 \text{ K}\]**Final Temperature:**\[P V_f = n R T_f\]\[T_f = \frac{P V_f}{n R} = \frac{(2.53 \times 10^5 \text{ Pa})(4.50 \times 10^{-2} \text{ m}^3)}{3 \times 8.314 \text{ J/mol K}} \approx 457.1 \text{ K}\]

Step 3: Calculate Work Done by the Gas

The work done by a gas at constant pressure during expansion is given by \(W = P\Delta V\), where \(\Delta V = V_f - V_i\).\[W = 2.53 \times 10^5 \text{ Pa} \times (4.50 \times 10^{-2} \text{ m}^3 - 3.20 \times 10^{-2} \text{ m}^3) = 3296.25 \text{ J}\]

Step 4: Calculate Heat Added to the Gas

For a monatomic ideal gas, the heat added can be found using \(Q = nC_p\Delta T\) where \(C_p = \frac{5}{2}R\).\[Q = 3 \times \frac{5}{2} \times 8.314 \times (457.1 - 325.3)\]\[Q \approx 1630.89 \times 131.8 = 214720.3 \text{ J}\]

Step 5: Calculate Change in Internal Energy

The change in internal energy \(\Delta U\) for a monatomic ideal gas can be calculated using \(\Delta U = nC_v\Delta T\) where \(C_v = \frac{3}{2}R\).\[\Delta U = 3 \times \frac{3}{2} \times 8.314 \times (457.1 - 325.3)\]\[\Delta U \approx 1171.41 \times 131.8 = 154720.2 \text{ J}\]

Step 6: Validate Energy Conservation

Check that the first law of thermodynamics \(\Delta U = Q - W\) holds:\[154720.2 \text{ J} \approx 214720.3 \text{ J} - 3296.25 \text{ J}\] Verify both sides of the equation are equal within reasonable rounding boundaries.

Key Concepts

Thermodynamics

Thermodynamics is the science related to the study of energy interactions, primarily focusing on heat and work transfers. In our scenario, we are dealing with an ideal monatomic gas undergoing expansion at constant pressure. This context involves understanding how energy is transferred and transformed in the system.
A key aspect of thermodynamics is the ideal gas law, which states that \(PV = nRT\), where:

• \(P\) is pressure in Pascals,
• \(V\) is volume in cubic meters,
• \(n\) is the number of moles,
• \(R\) is the universal gas constant (8.314 J/mol K), and
• \(T\) is the temperature in Kelvin.

The ideal gas law will help us compute the initial and final temperatures when the gas expands. Understanding these basic principles is crucial to solving problems involving gases under different conditions.

Work Done by Gas

When a gas expands, it can do work on its surroundings. For a process at constant pressure, like in this exercise, the work done by a gas can be calculated with the formula \(W = P \Delta V\), where \(\Delta V = V_f - V_i\) is the change in volume.
This equation tells us that:

• The work done (\(W\)) is positive when a gas expands because it pushes against the surrounding pressure.
• \(P\) is the constant external pressure applied to the system.
• The greater the change in volume, the more work is done.

In our specific case, knowing the initial and final volumes, along with the constant pressure, allows us to calculate the total work done during the gas expansion.

Change in Internal Energy

Internal energy (\(U\)) of a system is associated with the total kinetic energy possessed by the molecules of the system. For a monatomic ideal gas, the change in internal energy (\(\Delta U\)) is determined by the formula \(\Delta U = nC_v \Delta T\).
The key points are:

• \(C_v = \frac{3}{2}R\) is the specific heat at constant volume for a monatomic gas.
• \(\Delta T = T_f - T_i\) represents the change in temperature.

This relationship helps us see that any change in temperature directly affects the internal energy. Internal energy increases with an increase in temperature, meaning the gas molecules move faster and possess more kinetic energy.

Heat Transfer

In thermodynamics, heat transfer refers to the movement of thermal energy from one place to another. Using the equation \(Q = nC_p \Delta T\) for this solution, we assess the heat added to our monatomic ideal gas.
Here:

• \(Q\) is the heat added to the system,
• \(C_p = \frac{5}{2}R\) is the specific heat capacity at constant pressure for a monatomic gas, and
• \(\Delta T = T_f - T_i\) is the change in temperature.

The first law of thermodynamics states that the change in internal energy \(\Delta U\) is equal to the heat added to the system minus the work done by the system: \(\Delta U = Q - W\). Understanding this balance helps us evaluate how energy flows through the system, ensuring conservation of energy within the framework of thermodynamics.