A sample of a monatomic ideal gas is originally at `20^@ C`. What is the final temperature of the gas if both the pressure and volume are doubled?

Options

A. `5^@ C`
B. `20 ^@ C`
C. `80^@ C`
D. `900^@ C `

Answer

To solve the problem, we will use the Ideal Gas Law, which states:

\[ PV = nRT \]

Where:
- \( P \) = Pressure
- \( V \) = Volume
- \( n \) = Number of moles of gas (constant)
- \( R \) = Ideal gas constant (constant)
- \( T \) = Temperature in Kelvin

### Step-by-Step Solution:

1. **Convert the Initial Temperature to Kelvin**:
The initial temperature \( T_1 \) is given as \( 20^\circ C \). To convert this to Kelvin:
\[ T_1 = 20 + 273 = 293 \, K \]

**Hint**: Remember to always convert Celsius to Kelvin by adding 273.

2. **Identify the Initial and Final Conditions**:
- Initial Pressure \( P_1 = P \)
- Initial Volume \( V_1 = V \)
- Initial Temperature \( T_1 = 293 \, K \)

The problem states that both pressure and volume are doubled:
- Final Pressure \( P_2 = 2P \)
- Final Volume \( V_2 = 2V \)

**Hint**: Keep track of how the pressure and volume change; they both double in this case.

3. **Set Up the Ideal Gas Law for Initial and Final States**:
According to the Ideal Gas Law:
\[ P_1 V_1 = nRT_1 \quad \text{and} \quad P_2 V_2 = nRT_2 \]

Substituting the known values:
\[ PV = nR(293) \quad \text{and} \quad (2P)(2V) = nRT_2 \]

4. **Simplify the Final State Equation**:
The equation for the final state becomes:
\[ 4PV = nRT_2 \]

5. **Relate the Two Equations**:
Since \( PV = nR(293) \), we can substitute this into the equation for the final state:
\[ 4(nR(293)) = nRT_2 \]

This simplifies to:
\[ 4 \times 293 = T_2 \]

6. **Calculate the Final Temperature**:
\[ T_2 = 1172 \, K \]

**Hint**: When you multiply, ensure you keep track of units and calculations.

7. **Convert the Final Temperature Back to Celsius**:
To convert \( T_2 \) back to Celsius:
\[ T_2 = 1172 - 273 = 899 \, ^\circ C \]

**Hint**: To convert Kelvin back to Celsius, subtract 273.

### Final Answer:
The final temperature of the gas is approximately \( 899 \, ^\circ C \).