Find the molar specific heat of the process `p=a/T` for a monoatomic gas, a being constant.

Explanation

To find the molar specific heat of the process defined by \( P = \frac{a}{T} \) for a monoatomic gas, we can follow these steps:

### Step 1: Define Molar Specific Heat
The molar specific heat \( C \) is defined as the amount of heat added per unit temperature change. Mathematically, it can be expressed as:
\[ C = \frac{dQ}{dT} \]

### Step 2: Apply the First Law of Thermodynamics
According to the first law of thermodynamics:
\[ dU = dQ - dW \]
This can be rearranged to express \( dQ \):
\[ dQ = dU + dW \]

### Step 3: Express \( dU \) for a Monoatomic Gas
For a monoatomic ideal gas, the change in internal energy \( dU \) is given by:
\[ dU = nC_V dT \]
For one mole of gas (\( n = 1 \)), this simplifies to:
\[ dU = C_V dT \]

### Step 4: Express \( dW \)
The work done \( dW \) in a process can be expressed as:
\[ dW = P dV \]

### Step 5: Substitute \( dQ \) in Terms of \( dT \)
Substituting the expressions for \( dU \) and \( dW \) into the equation for \( dQ \):
\[ dQ = C_V dT + P dV \]

### Step 6: Differentiate with Respect to \( T \)
Now, we differentiate \( dQ \) with respect to \( T \):
\[ \frac{dQ}{dT} = C_V + P \frac{dV}{dT} \]
Thus, we have:
\[ C = C_V + P \frac{dV}{dT} \]

### Step 7: Relate \( P \) and \( V \) Using the Ideal Gas Law
Using the ideal gas law \( PV = nRT \), for one mole (\( n = 1 \)):
\[ P = \frac{RT}{V} \]
Given \( P = \frac{a}{T} \), we can set these equal:
\[ \frac{RT}{V} = \frac{a}{T} \]
From this, we can solve for \( V \):
\[ V = \frac{RT^2}{a} \]

### Step 8: Differentiate \( V \) with Respect to \( T \)
Now, we differentiate \( V \) with respect to \( T \):
\[ \frac{dV}{dT} = \frac{d}{dT} \left( \frac{RT^2}{a} \right) = \frac{2RT}{a} \]

### Step 9: Substitute \( \frac{dV}{dT} \) Back into the Equation for \( C \)
Substituting \( \frac{dV}{dT} \) back into the equation for \( C \):
\[ C = C_V + P \left( \frac{2RT}{a} \right) \]

### Step 10: Substitute \( P \) into the Equation
Substituting \( P = \frac{a}{T} \):
\[ C = C_V + \frac{a}{T} \cdot \frac{2RT}{a} \]
This simplifies to:
\[ C = C_V + 2R \]

### Step 11: Calculate \( C_V \) for a Monoatomic Gas
For a monoatomic gas, the molar specific heat at constant volume \( C_V \) is:
\[ C_V = \frac{3R}{2} \]

### Step 12: Final Calculation
Substituting \( C_V \) into the equation for \( C \):
\[ C = \frac{3R}{2} + 2R = \frac{3R}{2} + \frac{4R}{2} = \frac{7R}{2} \]

### Final Answer
Thus, the molar specific heat of the process \( P = \frac{a}{T} \) for a monoatomic gas is:
\[ C = \frac{7R}{2} \]