The ratio of P to V at any instant is constant and is equal to 1. for a monoatomic ideal gas under going a process. What is the molar heat capacity of the gas

Options

A. `(3R)/(2)`
B. `(4R)/(2)`
C. `(5R)/(2)`
D. 0

Answer

To find the molar heat capacity of a monoatomic ideal gas undergoing a process where the ratio of pressure (P) to volume (V) is constant and equal to 1, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Given Condition**:
The ratio of pressure to volume is constant and equal to 1:
\[ \frac{P}{V} = 1 \implies P = V \]

2. **Using the Ideal Gas Law**:
For 1 mole of an ideal gas, we have the ideal gas equation:
\[ PV = nRT \quad \text{(where } n = 1 \text{ for 1 mole)} \]
Therefore, we can write:
\[ PV = RT \]

3. **Differentiating the Ideal Gas Equation**:
We can express the differential form of the equation:
\[ d(PV) = PdV + VdP = d(RT) \]
Since \(R\) is a constant, we have:
\[ PdV + VdP = RdT \]

4. **Substituting P with V**:
From the earlier step, since \(P = V\), we can substitute \(P\) in the equation:
\[ VdV + VdP = RdT \]
This simplifies to:
\[ VdV + VdV = RdT \implies 2VdV = RdT \]

5. **Rearranging the Equation**:
Dividing both sides by 2:
\[ VdV = \frac{RdT}{2} \]

6. **Using the First Law of Thermodynamics**:
The first law of thermodynamics states:
\[ dQ = dU + dW \]
For an ideal gas, the change in internal energy \(dU\) for a monoatomic gas is given by:
\[ dU = C_V dT \]
The work done \(dW\) is:
\[ dW = PdV \]

7. **Substituting for dW**:
We can substitute \(PdV\) from our previous result:
\[ dW = VdV = \frac{RdT}{2} \]

8. **Combining the Equations**:
Now substituting \(dW\) into the first law:
\[ dQ = C_V dT + \frac{RdT}{2} \]

9. **Factoring out dT**:
Factoring out \(dT\):
\[ dQ = \left(C_V + \frac{R}{2}\right) dT \]

10. **Identifying Molar Heat Capacity**:
The molar heat capacity \(C\) is defined as:
\[ C = C_V + \frac{R}{2} \]
For a monoatomic ideal gas, \(C_V = \frac{3R}{2}\). Therefore:
\[ C = \frac{3R}{2} + \frac{R}{2} = \frac{4R}{2} = 2R \]

### Final Answer:
The molar heat capacity of the gas is:
\[ C = 2R \]