What is the heat input needed to raise the temperature of 2 moles of helium gas from `0^@C` to `100^@C`
(a) at constant volume,
(b) at constant pressure?
(c) what is the work done by the gas in part (b)?
Give your answer in terms of `R`.

Explanation

To solve the problem of finding the heat input needed to raise the temperature of 2 moles of helium gas from \(0^\circ C\) to \(100^\circ C\) at constant volume and constant pressure, as well as the work done by the gas, we can follow these steps:

### Step-by-Step Solution

**Given:**
- Number of moles, \(n = 2\)
- Initial temperature, \(T_1 = 0^\circ C = 273.15 \, K\)
- Final temperature, \(T_2 = 100^\circ C = 373.15 \, K\)
- Change in temperature, \(\Delta T = T_2 - T_1 = 100^\circ C = 100 \, K\)

**(a) Heat input at constant volume:**

1. **Identify the specific heat at constant volume (\(C_v\)) for helium:**
- Helium is a monoatomic gas, so:
\[ C_v = \frac{3}{2} R \]

2. **Calculate the heat input (\(Q\)) at constant volume using the formula:**
\[ Q = n C_v \Delta T \]
Substituting the values:
\[ Q = 2 \left(\frac{3}{2} R\right) (100) = 2 \cdot \frac{3}{2} R \cdot 100 = 300 R \]

**(b) Heat input at constant pressure:**

1. **Identify the specific heat at constant pressure (\(C_p\)) for helium:**
- For a monoatomic gas:
\[ C_p = \frac{5}{2} R \]

2. **Calculate the heat input (\(Q\)) at constant pressure:**
\[ Q = n C_p \Delta T \]
Substituting the values:
\[ Q = 2 \left(\frac{5}{2} R\right) (100) = 2 \cdot \frac{5}{2} R \cdot 100 = 500 R \]

**(c) Work done by the gas in part (b):**

1. **Use the first law of thermodynamics to find work done (\(W\)):**
\[ W = Q - \Delta U \]
Where \(\Delta U\) (change in internal energy) is given by:
\[ \Delta U = n C_v \Delta T \]

2. **Calculate \(\Delta U\):**
\[ \Delta U = 2 \left(\frac{3}{2} R\right) (100) = 300 R \]

3. **Substituting into the work done equation:**
\[ W = Q - \Delta U = 500 R - 300 R = 200 R \]

### Final Answers:
- (a) Heat input at constant volume: **\(300 R\)**
- (b) Heat input at constant pressure: **\(500 R\)**
- (c) Work done by the gas: **\(200 R\)**