For an ideal gas the molar heat capacity varies as `C=C_V+3aT^2`. Find the equation of the process in the variables (T,V) where a is a constant.

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the given information
We are given that the molar heat capacity \( C \) of an ideal gas varies as:
\[ C = C_V + 3aT^2 \]
where \( C_V \) is the molar heat capacity at constant volume, \( a \) is a constant, and \( T \) is the temperature.

### Step 2: Apply the First Law of Thermodynamics
According to the first law of thermodynamics:
\[ dq = du + dw \]
For one mole of an ideal gas, we can express \( dq \) as:
\[ dq = C dT \]
The change in internal energy \( du \) can be expressed as:
\[ du = C_V dT \]
And the work done \( dw \) can be expressed as:
\[ dw = P dV \]
Substituting these into the first law gives:
\[ C dT = C_V dT + P dV \]

### Step 3: Substitute the expression for C
Substituting \( C \) into the equation:
\[ (C_V + 3aT^2) dT = C_V dT + P dV \]
This simplifies to:
\[ 3aT^2 dT = P dV \]

### Step 4: Use the Ideal Gas Law
From the ideal gas law, we know:
\[ PV = nRT \]
For one mole of gas, this simplifies to:
\[ P = \frac{RT}{V} \]
Substituting this into our equation gives:
\[ 3aT^2 dT = \frac{RT}{V} dV \]

### Step 5: Rearrange the equation
Rearranging the equation, we have:
\[ 3aT^2 dT = \frac{RT}{V} dV \]
This can be rewritten as:
\[ \frac{3aT^2}{R} dT = \frac{dV}{V} \]

### Step 6: Integrate both sides
Now, we will integrate both sides:
\[ \int \frac{3aT^2}{R} dT = \int \frac{dV}{V} \]
The left side integrates to:
\[ \frac{3a}{R} \cdot \frac{T^3}{3} = \frac{aT^3}{R} \]
The right side integrates to:
\[ \ln V \]
Thus, we have:
\[ \frac{aT^3}{R} = \ln V + \ln D \]
where \( D \) is a constant of integration.

### Step 7: Simplify the equation
Using properties of logarithms, we can rewrite this as:
\[ \frac{aT^3}{R} = \ln (DV) \]
Exponentiating both sides gives:
\[ DV = e^{\frac{aT^3}{R}} \]
or
\[ V = \frac{e^{\frac{aT^3}{R}}}{D} \]

### Step 8: Final form of the equation
We can express this as:
\[ Ve^{-\frac{aT^3}{R}} = \text{constant} \]

### Conclusion
The equation of the process in the variables \( T \) and \( V \) is:
\[ V e^{-\frac{aT^3}{R}} = \text{constant} \]