If f denotes the degree of freedom of a gas, the ratio of two specific heats `(C_(P))/(C_(V))` is given by

Options

A. `(1)/(f)+1`
B. `(2)/(f)+1`
C. `(1)/(f)-1`
D. `(2)/(f)-1`

Explanation

To find the ratio of the specific heats \( \frac{C_P}{C_V} \) in terms of the degrees of freedom \( F \) of a gas, we can follow these steps:

### Step 1: Understand the Degrees of Freedom
The degrees of freedom \( F \) of a gas refers to the number of independent ways in which the gas molecules can move. For a monoatomic gas, \( F = 3 \) (translational motion), for a diatomic gas, \( F = 5 \) (3 translational + 2 rotational), and for more complex molecules, \( F \) can be higher.

### Step 2: Use the Equipartition Theorem
According to the equipartition theorem, the internal energy \( U \) of a gas is related to its degrees of freedom. The total internal energy can be expressed as:
\[ U = \frac{F}{2} nRT \]
where \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the temperature.

### Step 3: Relate Internal Energy to Heat Capacities
The heat capacity at constant volume \( C_V \) is defined as:
\[ C_V = \frac{dU}{dT} \]
From the expression for internal energy, we can differentiate with respect to temperature:
\[ C_V = \frac{d}{dT}\left(\frac{F}{2} nRT\right) = \frac{F}{2} nR \]

### Step 4: Use the Relation between \( C_P \) and \( C_V \)
For an ideal gas, the relationship between the heat capacities is given by:
\[ C_P - C_V = R \]
Substituting the expression for \( C_V \):
\[ C_P - \frac{F}{2} nR = R \]
This can be rearranged to find \( C_P \):
\[ C_P = \frac{F}{2} nR + R = \left(\frac{F}{2} + 1\right) nR \]

### Step 5: Find the Ratio \( \frac{C_P}{C_V} \)
Now we can find the ratio of the specific heats:
\[ \frac{C_P}{C_V} = \frac{\left(\frac{F}{2} + 1\right) nR}{\frac{F}{2} nR} \]
This simplifies to:
\[ \frac{C_P}{C_V} = \frac{\frac{F}{2} + 1}{\frac{F}{2}} = 1 + \frac{2}{F} \]

### Final Result
Thus, the ratio of the specific heats \( \frac{C_P}{C_V} \) is given by:
\[ \frac{C_P}{C_V} = 1 + \frac{2}{F} \]