The molar heat capacity of a monoatomic gas for which the ratio of pressure and volume is one.

Options

A. 4/2 R
B. 3/2 R
C. 5/2 R
D. zero

Answer

To find the molar heat capacity of a monoatomic gas for which the ratio of pressure to volume is one, we can follow these steps:

### Step 1: Understand the Given Ratio
We are given that the ratio of pressure (P) to volume (V) is 1. This can be expressed mathematically as:
\[ \frac{P}{V} = 1 \]
This implies:
\[ P = V \]

### Step 2: Relate Pressure, Volume, and Temperature
For an ideal gas, we can use the ideal gas law:
\[ PV = nRT \]
Since \( P = V \), we can substitute this into the ideal gas law:
\[ V^2 = nRT \]
From this, we can express the temperature as:
\[ T = \frac{V^2}{nR} \]

### Step 3: Determine the Value of Gamma (γ)
The ratio of specific heats, gamma (γ), is defined as:
\[ \gamma = \frac{C_p}{C_v} \]
For a monoatomic gas, we know that:
\[ C_v = \frac{3}{2}R \]
We need to find the value of \( C_p \) using the relationship:
\[ C_p = C_v + R \]
Thus,
\[ C_p = \frac{3}{2}R + R = \frac{5}{2}R \]
Now we can find γ:
\[ \gamma = \frac{C_p}{C_v} = \frac{\frac{5}{2}R}{\frac{3}{2}R} = \frac{5}{3} \]

### Step 4: Use the Formula for Molar Heat Capacity (C)
The molar heat capacity (C) can be expressed in terms of \( C_v \) and \( \gamma \):
\[ C = C_v + \frac{R}{1 - \gamma} \]
Substituting the values we have:
\[ C = \frac{3}{2}R + \frac{R}{1 - \frac{5}{3}} \]

### Step 5: Simplify the Expression
First, calculate \( 1 - \frac{5}{3} \):
\[ 1 - \frac{5}{3} = \frac{3}{3} - \frac{5}{3} = -\frac{2}{3} \]
Now substituting this back into the equation for C:
\[ C = \frac{3}{2}R + \frac{R}{-\frac{2}{3}} \]
\[ C = \frac{3}{2}R - \frac{3}{2}R = 0 \]
This indicates that the process is not valid under normal conditions.

### Final Answer
The molar heat capacity of the monoatomic gas under the given conditions cannot be determined as it leads to an invalid scenario.