Problem 27 - University Physics with Modern Physics

Question

CALC The temperature of 0.150 mol of an ideal gas is held constant at \(77.0^{\circ} \mathrm{C}\) while its volume is reduced to 25.0\(\%\) of its initial volume. The initial pressure of the gas is 1.25 atm. (a) Determine the work done by the gas. (b) What is the change in its internal energy? (c) Does the gas exchange heat with its surroundings? If so, how much? Does the gas absorb or liberate heat?

Step-by-Step Solution

Answer

(a) Work done, \(W = 646.5 \text{ J}\); (b) \(\Delta U = 0\); (c) Heat exchanged, \(Q = -646.5 \text{ J}\), the gas absorbs heat.

Step 1: Analyze the problem and gather information

To solve this problem, we need to first understand that the temperature is constant, meaning this is an isothermal process for an ideal gas. In such processes, the internal energy change is zero. We are given: \(n = 0.150\) mol, \(T = 77.0^{\circ} \mathrm{C} = 350.15 \mathrm{K}\), initial pressure \(P_i = 1.25 \text{ atm}\) and the volume changes to 25% of its initial value.

Step 2: Use the Ideal Gas Law

For isothermal processes, we can use the ideal gas law \(PV = nRT\) to determine the initial and final volumes and pressures (if needed). The change in volume isn't directly used in calculations for work done in isothermal processes, but helps check consistency.

Step 3: Calculate the work done

In an isothermal expansion or compression of an ideal gas, work done by the gas is given by the formula \(W = nRT\ln\frac{V_f}{V_i}\). Since it's compression, \(V_f = 0.25V_i\), so \(\ln\frac{V_f}{V_i} = \ln 0.25\). Substitute the known values to calculate \(W\).

Step 4: Determine the change in internal energy

For isothermal processes involving an ideal gas, the change in internal energy \(\Delta U = 0\) because the temperature remains constant.

Step 5: Assess if there is heat exchange and its quantity

Since \(\Delta U = Q - W\), and \(\Delta U = 0\), the heat exchanged with the surroundings, \(Q\), is equal to the work done on/by the system but with opposite sign, i.e., \(Q = W\). The gas either absorbs heat if \(W > 0\) (when compressed) or releases it otherwise.

Key Concepts

Isothermal Process

During an isothermal process, the temperature of the system remains constant. This is a special condition where the system, often an ideal gas, does not change its internal energy as it undergoes an expansion or compression. This occurs because any work done by the gas is exactly balanced by the heat exchanged with the surroundings.

In isothermal processes, the equation governing the ideal gas—called the Ideal Gas Law—remains valid, and the constant temperature implies constant internal energy. An ideal gas experiencing this process will maintain its energy state, as any input or removal of energy comes in the form of heat, which further facilitates the conditions for maintaining a steady temperature.

When addressing problems involving isothermal processes, it's critical to remember that while pressure and volume can change, these changes are synchronous in such a way that the product of pressure and volume remains a constant as dictated by the ideal gas law.

Internal Energy

The internal energy of a system refers to the total energy stored within it, encompassing translational, vibrational, and rotational energy of its particles. In the specific case of an ideal gas, which is often considered in physics and chemistry problems due to its simplified behavior, the internal energy relies exclusively on its temperature.

Isothermal processes bring an interesting characteristic: the internal energy remains unchanged. This happens because the temperature does not shift—keeping the energy levels of particles consistent. Therefore, for an ideal gas undergoing an isothermal process, the change in internal energy ( otates U&) is zero.

Understanding internal energy is crucial for thermodynamics as it underscores how energy is balanced in myriad processes, particularly in those that involve heat exchange and mechanical work such as expanding or compressing gases.

Work Done

Work done by or on a gas during an isothermal process is a pivotal concept. In such a process, since the temperature is constant thanks to heat exchange, the work done depends directly on the change in volume. For an isothermal compression or expansion, the work done (otates W&) is calculated using the formula: \[ W = nRT \ln \frac{V_f}{V_i} \] Where:

• \( n \): number of moles of gas
• \( R \): the universal gas constant
• \( T \): the absolute temperature, constant in isothermal processes
• \( V_f \) and \( V_i \): final and initial volumes, respectively

It's important to note the logarithmic relationship, which illustrates how the ratio of volumes—and indirectly, the pressure—determines the mechanical work done, always linked with heat changes in an ideal manner. This relationship enables us to determine whether work is done on the gas (compression) or by the gas (expansion).

Heat Exchange

Heat exchange, denoted as \( Q \), describes how energy in the form of heat is transferred between a system and its surroundings. During an isothermal process of an ideal gas, the heat exchange perfectly balances the work done by or on the system, ensuring a constant internal energy.

In mathematical terms, the first law of thermodynamics is applied: \[ \Delta U = Q - W \] Since \( \Delta U = 0 \) in an isothermal process, we infer that: \[ Q = W \] Here, \( Q \) is simply the work done but with the opposite sign, meaning energy entering the system as heat if compressed (making \( Q > 0 \)) or leaving if expanded (making \( Q < 0 \)). Understanding heat exchange is vital for any analysis involving temperature-dependent processes, ensuring proper energy balance and temperature control.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in thermodynamics that relates the pressure \( P \), volume \( V \), and temperature \( T \) of an ideal gas with its number of moles \( n \) and the universal gas constant \( R \). It is expressed as: \[ PV = nRT \] This law forms the backbone of analyzing gas behaviors under different conditions. In this context, especially during isothermal processes, it dictates that while the individual parameters like pressure and volume may change, their product remains constant when you hold temperature steady.

Key points of the Ideal Gas Law include:

• Explains how gases respond to changes in pressure, volume, and temperature
• Applies uniquely to ideal gases where interactions between molecules are negligible
• Enables prediction of one state property if the others (pressure, volume, or temperature) are known, assuming a fixed amount of gas

By mastering the Ideal Gas Law, you can navigate problems involving gas systems and understand other laws that extend this concept, expanding your toolkit in studying thermodynamics.