A sample of gas contracts by 1 litre against a constant pressure of 0.1 atm while 5.13 J heat it lost to surroundings. The change in internal energy, U of the system is

Options

A. 10.26 J
B. 5.0 J
C. 5.64 J
D. 4.0 J

Answer

To find the change in internal energy (ΔU) of the gas system, we will use the first law of thermodynamics, which states:

ΔU = q + w

Where:
- ΔU is the change in internal energy.
- q is the heat exchanged.
- w is the work done on or by the system.

### Step 1: Identify the heat exchanged (q)
The problem states that the gas loses 5.13 J of heat to the surroundings. Since this is a loss of heat, we will consider it as negative:

q = -5.13 J

**Hint:** Remember that heat lost by the system is negative, while heat gained is positive.

### Step 2: Calculate the work done (w)
The gas contracts by 1 liter against a constant pressure of 0.1 atm. The work done on the system (by the surroundings) can be calculated using the formula:

w = -P_external × ΔV

Here, ΔV (change in volume) is negative since the gas is contracting. Therefore, we have:

- P_external = 0.1 atm
- ΔV = -1 L (since the volume decreases)

Now substituting these values into the formula:

w = - (0.1 atm) × (-1 L)

To convert this to Joules, we use the conversion factor:
1 L·atm = 101.3 J

So,

w = 0.1 atm × 1 L × 101.3 J/L·atm
w = 10.13 J

**Hint:** Always remember to convert units when necessary, especially when dealing with energy.

### Step 3: Substitute q and w into the first law equation
Now we can substitute the values of q and w into the first law equation:

ΔU = q + w
ΔU = (-5.13 J) + (10.13 J)

### Step 4: Calculate ΔU
Now, performing the calculation:

ΔU = -5.13 J + 10.13 J
ΔU = 5.00 J

### Conclusion
The change in internal energy (ΔU) of the system is:

ΔU = 5.00 J

**Final Answer:** The change in internal energy (ΔU) of the system is 5.00 J.