In an adiabatic change the pressure and temperature of monoatomic gas are related as `P prop T^(c )` where C equal

Options

A. `2//5`
B. `5//2`
C. `3//5`
D. `5//3 `

Explanation

To solve the problem, we need to establish the relationship between pressure (P) and temperature (T) for a monoatomic gas undergoing an adiabatic process. The steps are as follows:

### Step 1: Understand the Adiabatic Process
In an adiabatic process, there is no heat exchange with the surroundings. For an ideal gas, the relationship between pressure (P), volume (V), and temperature (T) can be described using the equation:
\[ PV^{\gamma} = \text{constant} \]
where \(\gamma\) (gamma) is the heat capacity ratio \(C_p/C_v\).

### Step 2: Express Volume in Terms of Pressure and Temperature
Using the ideal gas law:
\[ PV = nRT \]
we can express volume (V) as:
\[ V = \frac{nRT}{P} \]

### Step 3: Substitute Volume into the Adiabatic Equation
Substituting the expression for V into the adiabatic equation:
\[ P \left(\frac{nRT}{P}\right)^{\gamma} = \text{constant} \]
This simplifies to:
\[ P^{1 - \gamma} (nRT)^{\gamma} = \text{constant} \]

### Step 4: Rearranging the Equation
Rearranging gives us:
\[ P^{1 - \gamma} \propto (nRT)^{\gamma} \]
This implies:
\[ P^{1 - \gamma} \propto T^{\gamma} \]

### Step 5: Expressing Pressure in Terms of Temperature
From the proportional relationship, we can express pressure as:
\[ P \propto T^{\frac{\gamma}{1 - \gamma}} \]

### Step 6: Identifying the Value of C
Comparing this with the given relationship \(P \propto T^C\), we find that:
\[ C = \frac{\gamma}{1 - \gamma} \]

### Step 7: Finding the Value of Gamma for Monoatomic Gas
For a monoatomic gas, the value of \(\gamma\) is:
\[ \gamma = \frac{C_p}{C_v} = \frac{5}{3} \]

### Step 8: Substituting Gamma into the Expression for C
Substituting \(\gamma\) into the expression for C:
\[ C = \frac{\frac{5}{3}}{1 - \frac{5}{3}} = \frac{\frac{5}{3}}{-\frac{2}{3}} = -\frac{5}{2} \]

### Step 9: Finalizing the Value of C
Thus, we find:
\[ C = \frac{5}{2} \]

### Conclusion
The value of C for the relationship between pressure and temperature in an adiabatic process for a monoatomic gas is:
\[ C = \frac{5}{2} \]