Question

Each molecule of a gas has \(\mathrm{f}\) degrees of freedom. The radio \(\left(\mathrm{C}_{\mathrm{p}} / \mathrm{C}_{\mathrm{v}}\right)=\gamma\) for the gas is (A) \(1+(\mathrm{f} / 2)\) (B) \(1+(1 / \mathrm{f})\) (C) \(1+(2 / \mathrm{f})\) (D) \(1+\\{(\mathrm{f}-1) / 3\\}\)

Short answer

The correct option for the ratio \(\left(\mathrm{C}_{\mathrm{p}} / \mathrm{C}_{\mathrm{v}}\right)=\gamma\) for the gas is (C) \(1+(2 / \mathrm{f})\).

Step-by-step solution

Relationship between specific heat capacities and degrees of freedom

To find the relationship between the specific heat capacities and the degrees of freedom (\(f\)), we can recall that: \(C_p=C_v+nR\) Where \(C_p\) = specific heat capacity at constant pressure, \(C_v\) = specific heat capacity at constant volume, \(n\) = number of moles, and \(R\) = ideal gas constant.

Equation of specific heat capacity at constant volume

For a gas with \(f\) degrees of freedom, the specific heat capacity at constant volume (\(C_v\)) is given by: \(C_v=\frac{f}{2}nR\) We are going to use this equation to express \(\frac{C_p}{C_v}\) in terms of \(f\).

Derive the expression for gamma (\(\gamma\))

To find the ratio \(\gamma\), we can substitute the expression of \(C_v\) (from step 2) into the equation from step 1: \(C_p - C_v = nR\) Then divide both sides by \(C_v\): \(\frac{C_p}{C_v}-1 = \frac{2}{f}\) Add 1 on both sides of the equation to get the ratio: \(\frac{C_p}{C_v} = \gamma = 1 + \frac{2}{f}\) Comparing our derived expression with the given options, we find that it matches with option (C).

Answer

The correct option for the ratio \(\left(\mathrm{C}_{\mathrm{p}} / \mathrm{C}_{\mathrm{v}}\right)=\gamma\) for the gas is (C) \(1+(2 / \mathrm{f})\)

Key concepts

Specific Heat Capacity

Specific heat capacity is an important concept in thermodynamics that measures how much heat energy a substance can absorb. It is the amount of heat required to raise the temperature of a unit mass of a substance by one degree Celsius.

• There are two main types of specific heat capacities for gases: at constant volume (\(C_v\)) and at constant pressure (\(C_p\)).
• Specific heat capacity at constant volume (\(C_v\)) refers to the amount of heat required to raise the temperature of a unit mass of gas by one degree while keeping the volume constant.
• Specific heat capacity at constant pressure (\(C_p\)) accounts for the same but keeps pressure constant.

These two values can differ because when gas is heated at constant pressure, it also does work by expanding, requiring more energy.
This is why \(C_p\) is always greater than \(C_v\). Understanding these concepts helps in calculating energy changes in thermodynamic processes.

Ideal Gas Constant

The ideal gas constant, often denoted as \(R\), is an essential factor in the equation of state for ideal gases. It relates pressure, volume, and temperature in a given quantity of gas.

• The ideal gas law is expressed as \(PV = nRT\), where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, and \(T\) is the temperature in Kelvin.
• The ideal gas constant \(R\) comes with a unit of \(J/(mol \, K)\) and has an approximate value of 8.314.

This constant provides a bridge between microscopic and macroscopic physical properties. It allows us to convert the number of particles into moles, which makes calculations involving temperature and pressure more consistent and comprehensible.

Specific Heat Ratio

The specific heat ratio, denoted by \(\gamma\), is a vital concept particularly in thermodynamics concerning gases. It is defined as the ratio of the specific heat capacity at constant pressure to that at constant volume. \(\frac{C_p}{C_v} = \gamma\).

• This ratio is crucial because it influences the speed of sound in the gas, as well as other thermal properties.
• In the context of degrees of freedom, the specific heat ratio can be calculated using \(C_v = \frac{f}{2}nR\) and the relationship \(C_p - C_v = nR\).

Applying these equations, we derive that \(\gamma = 1 + \frac{2}{f}\), where \(f\) is the degrees of freedom.
Understanding this ratio helps in predicting how a gas will behave under various thermodynamic conditions, and it often serves as a key parameter in engineering applications such as designing engines and turbines.