The internal energy of a gas is given by `U=1.5pV`. It expands from `100 cm^(3) to 200 cm^(3)` against a constant pressure of `1.0xx10^(5)` Pa. calculate the heat absorbed by the gas in the process.

Explanation

To solve the problem of calculating the heat absorbed by the gas during its expansion, we can follow these steps:

### Step 1: Understand the Given Information
- The internal energy of the gas is given by the equation \( U = 1.5 pV \).
- The gas expands from an initial volume \( V_1 = 100 \, \text{cm}^3 \) to a final volume \( V_2 = 200 \, \text{cm}^3 \).
- The pressure during the expansion is constant at \( p = 1.0 \times 10^5 \, \text{Pa} \).

### Step 2: Convert Volumes to Cubic Meters
Since the pressure is given in Pascals (which is in SI units), we need to convert the volumes from cubic centimeters to cubic meters:
\[ V_1 = 100 \, \text{cm}^3 = 100 \times 10^{-6} \, \text{m}^3 = 1.0 \times 10^{-4} \, \text{m}^3 \]
\[ V_2 = 200 \, \text{cm}^3 = 200 \times 10^{-6} \, \text{m}^3 = 2.0 \times 10^{-4} \, \text{m}^3 \]

### Step 3: Calculate the Work Done by the Gas
The work done \( W \) by the gas during an isobaric (constant pressure) process is given by:
\[ W = P \Delta V \]
where \( \Delta V = V_2 - V_1 \):
\[ \Delta V = 2.0 \times 10^{-4} \, \text{m}^3 - 1.0 \times 10^{-4} \, \text{m}^3 = 1.0 \times 10^{-4} \, \text{m}^3 \]
Now, substituting the values:
\[ W = (1.0 \times 10^5 \, \text{Pa}) \times (1.0 \times 10^{-4} \, \text{m}^3) = 10 \, \text{J} \]

### Step 4: Calculate the Change in Internal Energy
The change in internal energy \( \Delta U \) is given by:
\[ \Delta U = U_2 - U_1 = 1.5 p V_2 - 1.5 p V_1 \]
Factoring out \( 1.5p \):
\[ \Delta U = 1.5 p (V_2 - V_1) \]
Substituting the values:
\[ \Delta U = 1.5 \times (1.0 \times 10^5 \, \text{Pa}) \times (1.0 \times 10^{-4} \, \text{m}^3) = 1.5 \times 10 \, \text{J} = 15 \, \text{J} \]

### Step 5: Apply the First Law of Thermodynamics
According to the first law of thermodynamics:
\[ Q = W + \Delta U \]
Substituting the values we calculated:
\[ Q = 10 \, \text{J} + 15 \, \text{J} = 25 \, \text{J} \]

### Final Answer
The heat absorbed by the gas during the process is \( Q = 25 \, \text{J} \).

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