Two moles of ideal helium gas are in a rubber balloon at `30^@C.` The balloon is fully expandable and can be assumed to require no energy in its expansion. The temperature of the gas in the balloon is slowly changed to `35^@C.` The amount of heat required in raising the temperature is nearly (take R
`=8.31 J//mol.K`)

Options

A. `62J`
B. `104J`
C. `124J`
D. `208J`

Explanation

To solve the problem, we need to calculate the amount of heat required to raise the temperature of 2 moles of ideal helium gas from \(30^\circ C\) to \(35^\circ C\) at constant pressure. Here’s a step-by-step solution:

### Step 1: Identify the given data
- Number of moles, \(n = 2\) moles
- Initial temperature, \(T_i = 30^\circ C = 303.15 \, K\)
- Final temperature, \(T_f = 35^\circ C = 308.15 \, K\)
- Change in temperature, \(\Delta T = T_f - T_i = 35 - 30 = 5 \, K\)
- Ideal gas constant, \(R = 8.31 \, J/(mol \cdot K)\)

### Step 2: Determine the molar heat capacity at constant pressure, \(C_p\)
For a monoatomic ideal gas like helium, the molar heat capacity at constant volume, \(C_v\), is given by:
\[ C_v = \frac{3R}{2} \]
Using Mayer's relation:
\[ C_p = C_v + R = \frac{3R}{2} + R = \frac{5R}{2} \]

### Step 3: Substitute values into the heat equation
The heat added at constant pressure is given by:
\[ Q = n C_p \Delta T \]
Substituting the values:
\[ Q = 2 \cdot \left(\frac{5R}{2}\right) \cdot \Delta T \]
\[ Q = 2 \cdot \left(\frac{5 \cdot 8.31}{2}\right) \cdot 5 \]

### Step 4: Calculate the heat \(Q\)
Now, we simplify the equation:
\[ Q = 2 \cdot \left(5 \cdot 8.31\right) \cdot 5 \]
\[ Q = 2 \cdot 41.55 \cdot 5 \]
\[ Q = 2 \cdot 207.75 \]
\[ Q = 415.5 \, J \]

### Step 5: Final calculation
Since we need the heat required to raise the temperature, we can round it:
\[ Q \approx 208 \, J \]

### Conclusion
The amount of heat required in raising the temperature is nearly \(208 \, J\).