One cubic metre of an ideal gas is at a pressure of `10 N//m^2` and temperature 300K. The gas is allowed to expand at constant pressure to twice its volume by supplying heat. If the change in internal energy in this process is `10^4 `J, then the heat supplied is_

Options

A. `10^5J`
B. `10^4J`
C. `11 xx 10^4J`
D. `2.2 xx 10^5 J`

Answer

To solve the problem step by step, we will use the first law of thermodynamics, which states that the change in internal energy (ΔU) of a system is equal to the heat added to the system (Q) minus the work done by the system (W):

\[ \Delta U = Q - W \]

From this, we can rearrange the equation to find the heat supplied:

\[ Q = \Delta U + W \]

### Step 1: Identify the given values
- Initial volume (Vi) = 1 m³
- Final volume (Vf) = 2 Vi = 2 m³
- Pressure (P) = 10 N/m²
- Change in internal energy (ΔU) = \(10^4\) J

### Step 2: Calculate the change in volume (ΔV)
\[ \Delta V = V_f - V_i = 2 \, \text{m}^3 - 1 \, \text{m}^3 = 1 \, \text{m}^3 \]

### Step 3: Calculate the work done (W)
The work done by the gas during expansion at constant pressure is given by the formula:
\[ W = P \Delta V \]
Substituting the values:
\[ W = 10 \, \text{N/m}^2 \times 1 \, \text{m}^3 = 10 \, \text{J} \]

### Step 4: Calculate the heat supplied (Q)
Now we can substitute ΔU and W into the equation for Q:
\[ Q = \Delta U + W \]
Substituting the known values:
\[ Q = 10^4 \, \text{J} + 10 \, \text{J} = 10^4 \, \text{J} + 0.01 \times 10^4 \, \text{J} = 10^4 \, \text{J} (approximately) \]

### Final Answer
The heat supplied is approximately \(10^4\) J.

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