Question

A polyatomic ideal gas occupies an initial volume \(V_{\mathrm{i}}\), which is decreased to \(\frac{1}{8} V_{\mathrm{i}}\) via an adiabatic process. Which equation expresses the relationship between the initial and final temperatures of this gas? a) \(T_{\mathrm{f}}=\frac{2}{3} T_{\mathrm{i}}\) d) \(T_{\mathrm{f}}=\frac{9}{5} T_{\mathrm{i}}\) b) \(T_{\mathrm{f}}=2 T_{\mathrm{i}}\) e) \(T_{\mathrm{f}}=8 T_{\mathrm{i}}\) c) \(T_{\mathrm{f}}=16 T_{\mathrm{i}}\)

Short answer

Answer: The final temperature is related to the initial temperature by the formula: \(T_\mathrm{f}=\frac{2}{3} T_\mathrm{i}\).

Step-by-step solution

Recall the adiabatic condition for ideal gases

For an adiabatic process (no heat transfer), the equation relating the initial and final pressure, volume, and temperatures of an ideal gas is given by: \(PV^{\gamma} = \mathrm{constant}\), where \(P\) - Pressure of the gas, \(V\) - Volume of the gas, \(\gamma\) - Adiabatic index (the ratio of specific heat capacities of the gas, \(C_p/C_v\)).

Changes in initial and final volumes

The initial volume of the gas is given as \(V_\mathrm{i}\). The final volume is reduced to \(\frac{1}{8}V_\mathrm{i}\). We will use \(V_\mathrm{f}\) to denote the final volume.

Write the adiabatic condition for initial and final states

For the initial state, the adiabatic condition is: \(P_\mathrm{i} V_\mathrm{i}^{\gamma} = \mathrm{constant}\). For the final state, the adiabatic condition is: \(P_\mathrm{f} V_\mathrm{f}^{\gamma} = \mathrm{constant}\). Since both represent the same constant, we can equate the two expressions: \(P_\mathrm{i} V_\mathrm{i}^{\gamma} = P_\mathrm{f} V_\mathrm{f}^{\gamma}\).

Relate pressures using the ideal gas law

Recall the ideal gas law: \(PV = nRT\), where \(n\) is the number of moles, \(R\) is the gas constant, and \(T\) is the temperature. Initial state: \(P_\mathrm{i} V_\mathrm{i} = n R T_\mathrm{i}\), Final state: \(P_\mathrm{f} V_\mathrm{f} = n R T_\mathrm{f}\). We can now divide the final state equation by the initial state equation to get: \(\frac{P_\mathrm{f}}{P_\mathrm{i}} = \frac{T_\mathrm{f} V_\mathrm{f}}{T_\mathrm{i} V_\mathrm{i}}\).

Substitute the expressions for volume and pressure

Substitute \(V_\mathrm{f} = \frac{1}{8} V_\mathrm{i}\) and \(\frac{P_\mathrm{f}}{P_\mathrm{i}}\) from Step 4 into the equation from Step 3: \(P_\mathrm{i} V_\mathrm{i}^{\gamma} = P_\mathrm{f}(\frac{1}{8}V_\mathrm{i})^{\gamma}\). Cancel out \(V_\mathrm{i}^{\gamma}\) from both sides: \(P_\mathrm{i} = P_\mathrm{f}(\frac{1}{8})^{\gamma}\). Now, substitute this back into the equation from Step 4: \((\frac{1}{8})^{\gamma} = \frac{T_\mathrm{f}(\frac{1}{8})}{T_\mathrm{i}}\).

Solve for the final temperature in terms of the initial temperature

Now we need to find \(T_\mathrm{f}\) in terms of \(T_\mathrm{i}\): \((\frac{1}{8})^{\gamma} = \frac{T_\mathrm{f}}{8T_\mathrm{i}}\). Multiply both sides by \(8T_\mathrm{i}\): \(8T_\mathrm{i}(\frac{1}{8})^{\gamma} = T_\mathrm{f}\). For a polyatomic ideal gas, the adiabatic index \(\gamma = \frac{5}{3}\). Substitute this value: \(8T_\mathrm{i}(\frac{1}{8})^{\frac{5}{3}} = T_\mathrm{f}\). Evaluate the power: \(T_\mathrm{f} = 8^{\frac{2}{3}}T_\mathrm{i}\). Therefore, the correct answer is: \(T_\mathrm{f}=\frac{2}{3} T_\mathrm{i}\).

Key concepts

Polyatomic Ideal Gas

When we refer to a polyatomic ideal gas, we're describing a gas that consists of molecules with more than two atoms, yet it behaves according to the ideal gas model.

This model assumes that the gas molecules are point particles that do not interact with each other except during elastic collisions. In reality, this is somewhat of an approximation because real gases do have intermolecular forces and occupy space. However, the ideal gas model is very useful for understanding the behavior of gases under many conditions.

For a polyatomic ideal gas, the degrees of freedom are higher compared to monoatomic or diatomic gases due to their complex molecular structure. This affects the way energy is distributed within the gas. These additional degrees of freedom mean that polyatomic gases have higher specific heat capacities, which in turn influences processes such as adiabatic expansion or compression.

Adiabatic Index

The adiabatic index, denoted by the Greek letter \(\gamma\), is crucial to understanding adiabatic processes.

It is defined as the ratio of the specific heat capacity at constant pressure (\(C_p\)) to the specific heat capacity at constant volume (\(C_v\)). In essence, the adiabatic index is a measure of how much more heat capacity a gas has when it is allowed to expand and do work on its surroundings compared to when it's confined to a constant volume.

Note that for a polyatomic ideal gas, the adiabatic index is greater than those for monoatomic or diatomic gases as it has more degrees of freedom. This influences how the gas's temperature and volume will change during an adiabatic process.

Specific Heat Capacities

The specific heat capacities of a substance relate to the amount of heat required to raise the temperature of a unit mass of the substance by one degree.

There are two specific heat capacities crucial for thermodynamics: \(C_p\), the specific heat at constant pressure, and \(C_v\), the specific heat at constant volume. For gases, these values differ because holding a gas at constant volume while heating it doesn't allow it to do work on its surroundings, whereas at constant pressure, the gas can expand and thus requires more heat to achieve the same temperature rise.

The higher degrees of freedom seen in polyatomic gases result in higher specific heat capacities. Knowing \(C_p\) and \(C_v\) allows us to calculate the adiabatic index (\(\gamma = \frac{C_p}{C_v}\)) crucial for predicting behavior during adiabatic processes.

Ideal Gas Law

The ideal gas law is a fundamental equation that relates the pressure, volume, number of moles, and temperature of an ideal gas. The law is typically written as: \(PV = nRT\).

In this equation, \(P\) represents pressure, \(V\) is volume, \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(T\) signifies temperature. It's a valuable equation because it can be used to compute any one of the variables if the other three are known.

In the context of adiabatic processes for polyatomic ideal gases, the ideal gas law can be combined with the concept of the adiabatic process to predict how a gas's temperature will change as its volume changes without heat exchange with its surroundings.