Pressure of `1` mole ideal is given by
`P=P_(0)[1-(1)/(2)(V_(0)/(V))^(2)]` ,brgt If volume of gas change from `V_(0)` to `2 V_(0)`. Find change in temperature.

Options

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

Answer

To solve the problem, we need to find the change in temperature of 1 mole of an ideal gas when its volume changes from \( V_0 \) to \( 2V_0 \). The pressure of the gas is given by:

\[ P = P_0 \left( 1 - \frac{1}{2} \left( \frac{V_0}{V} \right)^2 \right) \]

### Step 1: Use the Ideal Gas Law
We start with the ideal gas equation:

\[ PV = nRT \]

For 1 mole of gas, \( n = 1 \). Thus, we can write:

\[ P = \frac{RT}{V} \]

### Step 2: Substitute Pressure in Terms of Temperature
We can substitute the expression for pressure \( P \) from the given equation into the ideal gas law:

\[ \frac{RT}{V} = P_0 \left( 1 - \frac{1}{2} \left( \frac{V_0}{V} \right)^2 \right) \]

### Step 3: Rearranging the Equation
Now, we can rearrange this equation to express \( T \) in terms of \( V \):

\[ RT = P_0 V \left( 1 - \frac{1}{2} \left( \frac{V_0}{V} \right)^2 \right) \]

Dividing both sides by \( R \):

\[ T = \frac{P_0 V}{R} \left( 1 - \frac{1}{2} \left( \frac{V_0}{V} \right)^2 \right) \]

### Step 4: Calculate Temperature at Initial Volume \( V_0 \)
Now, we calculate the temperature \( T_1 \) when \( V = V_0 \):

\[ T_1 = \frac{P_0 V_0}{R} \left( 1 - \frac{1}{2} \left( \frac{V_0}{V_0} \right)^2 \right) \]

This simplifies to:

\[ T_1 = \frac{P_0 V_0}{R} \left( 1 - \frac{1}{2} \right) = \frac{P_0 V_0}{R} \cdot \frac{1}{2} = \frac{P_0 V_0}{2R} \]

### Step 5: Calculate Temperature at Final Volume \( 2V_0 \)
Next, we calculate the temperature \( T_2 \) when \( V = 2V_0 \):

\[ T_2 = \frac{P_0 (2V_0)}{R} \left( 1 - \frac{1}{2} \left( \frac{V_0}{2V_0} \right)^2 \right) \]

This simplifies to:

\[ T_2 = \frac{P_0 (2V_0)}{R} \left( 1 - \frac{1}{2} \cdot \frac{1}{4} \right) = \frac{P_0 (2V_0)}{R} \left( 1 - \frac{1}{8} \right) = \frac{P_0 (2V_0)}{R} \cdot \frac{7}{8} \]

Thus,

\[ T_2 = \frac{7P_0 V_0}{4R} \]

### Step 6: Find the Change in Temperature
Now, we find the change in temperature \( \Delta T \):

\[ \Delta T = T_2 - T_1 = \frac{7P_0 V_0}{4R} - \frac{P_0 V_0}{2R} \]

To combine these fractions, we convert \( \frac{P_0 V_0}{2R} \) to have a common denominator:

\[ \Delta T = \frac{7P_0 V_0}{4R} - \frac{2P_0 V_0}{4R} = \frac{(7 - 2)P_0 V_0}{4R} = \frac{5P_0 V_0}{4R} \]

### Final Answer
Thus, the change in temperature is:

\[ \Delta T = \frac{5P_0 V_0}{4R} \]