One mole of an ideal monoatomic gas is heated at a constant pressure of one atmosphere from`0^(@)` to `100^(@)C.` Then the change in the internal energy is

Options

A. `6.56`joules
B. `8.32xx10^(2)`joules
C. `12.48xx10^(2)`joules
D. `20.80joules`

Answer

To solve the problem of finding the change in internal energy of one mole of an ideal monoatomic gas heated at constant pressure from \(0^\circ C\) to \(100^\circ C\), we can follow these steps:

### Step 1: Understand the relationship between heat, work, and internal energy
According to the first law of thermodynamics, we have:
\[ \Delta Q = \Delta U + \Delta W \]
where:
- \(\Delta Q\) is the heat added to the system,
- \(\Delta U\) is the change in internal energy,
- \(\Delta W\) is the work done by the system.

### Step 2: Identify the process
In this case, the gas is heated at constant pressure. For a constant pressure process, the work done by the gas can be expressed as:
\[ \Delta W = P \Delta V \]
where \(P\) is the pressure and \(\Delta V\) is the change in volume.

### Step 3: Use the specific heat at constant pressure
For an ideal gas, the heat added at constant pressure can be expressed as:
\[ \Delta Q = n C_p \Delta T \]
where:
- \(n\) is the number of moles,
- \(C_p\) is the specific heat at constant pressure,
- \(\Delta T\) is the change in temperature.

### Step 4: Calculate \(\Delta T\)
The change in temperature from \(0^\circ C\) to \(100^\circ C\) is:
\[ \Delta T = 100 - 0 = 100 \, \text{°C} = 100 \, \text{K} \]

### Step 5: Determine \(C_p\) for a monoatomic gas
For a monoatomic ideal gas, the specific heat at constant pressure is given by:
\[ C_p = \frac{5}{2} R \]
where \(R\) is the universal gas constant (\(R \approx 8.31 \, \text{J/(mol·K)}\)).

### Step 6: Calculate \(\Delta Q\)
Substituting the values into the heat equation:
\[ \Delta Q = n C_p \Delta T = 1 \cdot \frac{5}{2} R \cdot 100 \]
\[ \Delta Q = 1 \cdot \frac{5}{2} \cdot 8.31 \cdot 100 \]
\[ \Delta Q = 5 \cdot 8.31 \cdot 50 \]
\[ \Delta Q = 2077.5 \, \text{J} \]

### Step 7: Calculate the change in internal energy
For an ideal gas, the change in internal energy is related to the change in temperature:
\[ \Delta U = n C_v \Delta T \]
where \(C_v\) for a monoatomic gas is:
\[ C_v = \frac{3}{2} R \]
Thus,
\[ \Delta U = 1 \cdot \frac{3}{2} R \cdot 100 \]
\[ \Delta U = 1 \cdot \frac{3}{2} \cdot 8.31 \cdot 100 \]
\[ \Delta U = 3 \cdot 8.31 \cdot 50 \]
\[ \Delta U = 1246.5 \, \text{J} \]

### Final Answer
The change in internal energy is:
\[ \Delta U = 1246.5 \, \text{J} \approx 1.25 \times 10^3 \, \text{J} \]