A 1.0 kg bar of copper is heated at atmospheric pressure `(1.01xx10^5N//m^2)`. If its temperature increases from `20^@C` to `50^@C`, calculate the change in its internal energy. `alpha=7.0xx10^-6//^@C`, `rho=8.92xx10^3kg//m^3` and `c=387J//kg-^@C`.

Explanation

To calculate the change in internal energy of a 1.0 kg bar of copper heated from 20°C to 50°C, we will follow these steps:

### Step 1: Calculate the Volume of the Copper Bar
The volume \( V \) of the copper bar can be calculated using the formula:
\[ V = \frac{m}{\rho} \]
Where:
- \( m = 1.0 \, \text{kg} \) (mass of the copper bar)
- \( \rho = 8.92 \times 10^3 \, \text{kg/m}^3 \) (density of copper)

Substituting the values:
\[ V = \frac{1.0 \, \text{kg}}{8.92 \times 10^3 \, \text{kg/m}^3} = 1.12 \times 10^{-4} \, \text{m}^3 \]

### Step 2: Calculate the Coefficient of Volume Expansion
The coefficient of volume expansion \( \gamma \) is related to the coefficient of linear expansion \( \alpha \) by the formula:
\[ \gamma = 3\alpha \]
Given \( \alpha = 7.0 \times 10^{-6} \, \text{°C}^{-1} \):
\[ \gamma = 3 \times 7.0 \times 10^{-6} = 2.1 \times 10^{-5} \, \text{°C}^{-1} \]

### Step 3: Calculate the Change in Volume
The change in volume \( \Delta V \) can be calculated using the formula:
\[ \Delta V = V \cdot \gamma \cdot \Delta T \]
Where:
- \( \Delta T = 50°C - 20°C = 30°C \)

Substituting the values:
\[ \Delta V = 1.12 \times 10^{-4} \, \text{m}^3 \cdot 2.1 \times 10^{-5} \, \text{°C}^{-1} \cdot 30 \, \text{°C} = 7.056 \times 10^{-8} \, \text{m}^3 \]

### Step 4: Calculate the Work Done
The work done \( W \) at constant pressure can be calculated using:
\[ W = P \cdot \Delta V \]
Where:
- \( P = 1.01 \times 10^5 \, \text{N/m}^2 \)

Substituting the values:
\[ W = 1.01 \times 10^5 \, \text{N/m}^2 \cdot 7.056 \times 10^{-8} \, \text{m}^3 = 7.13 \times 10^{-3} \, \text{J} \]

### Step 5: Calculate the Heat Added
The heat \( Q \) added can be calculated using:
\[ Q = m \cdot c \cdot \Delta T \]
Where:
- \( c = 387 \, \text{J/(kg°C)} \)

Substituting the values:
\[ Q = 1.0 \, \text{kg} \cdot 387 \, \text{J/(kg°C)} \cdot 30 \, \text{°C} = 11610 \, \text{J} \]

### Step 6: Calculate the Change in Internal Energy
According to the first law of thermodynamics:
\[ \Delta U = Q - W \]
Substituting the values:
\[ \Delta U = 11610 \, \text{J} - 7.13 \times 10^{-3} \, \text{J} \approx 11609.99287 \, \text{J} \]

### Final Answer
The change in internal energy of the copper bar is approximately:
\[ \Delta U \approx 11610 \, \text{J} \]