Describe the isothermal process. A right rectangular prism of cast iron has dimensions of 55 × 45 × 35 mm at 25 °C. The prism is heated to 88 °C. The coefficient of linear expansion of cast iron is 9 × 10⁻⁶/°C. Calculate the following: 7.2.1 The increase in volume 7.2.2 The volume at 88 °C

Here are the solutions to Question 7:

QUESTION 7: HEAT

7.1 Describe the isothermal process.

An isothermal process is a thermodynamic process during which the temperature of a system remains constant. This occurs when the system is in thermal contact with a large heat reservoir, allowing heat to be exchanged to maintain a steady temperature.

7.2 A right rectangular prism of cast iron has dimensions of $55 \times 45 \times 35 \text{ mm}$ at $25^\circ \text{C}$. The prism is heated to $88^\circ \text{C}$. The coefficient of linear expansion of cast iron is $9 \times 10^{-6}/^\circ \text{C}$.

Given:

• Initial dimensions: $L_0 = 55 \text{ mm}$, $W_0 = 45 \text{ mm}$, $H_0 = 35 \text{ mm}$
• Initial temperature, $T_1 = 25^\circ \text{C}$
• Final temperature, $T_2 = 88^\circ \text{C}$
• Coefficient of linear expansion, $\alpha = 9 \times 10^{-6}/^\circ \text{C}$

7.2.1 The increase in volume

Step 1: Calculate the original volume ($V_0$). $V_0 = L_0 \times W_0 \times H_0$ $V_0 = 55 mm \times 45 mm \times 35 mm$ $V_0 = 86625 mm^3$

Step 2: Calculate the change in temperature ($\Delta T$). $\Delta T = T_2 - T_1$ $\Delta T = 88^\circ C - 25^\circ C = 63^\circ C$

Step 3: Calculate the coefficient of volumetric expansion ($\gamma$). For isotropic materials, the coefficient of volumetric expansion is approximately three times the coefficient of linear expansion. $\gamma = 3\alpha$ $\gamma = 3 \times (9 \times 10^{-6}/^\circ C) = 27 \times 10^{-6}/^\circ C$

Step 4: Calculate the increase in volume ($\Delta V$). $\Delta V = V_0 \gamma \Delta T$ $\Delta V = (86625 mm^3) \times (27 \times 10^{-6}/^\circ C) \times (63^\circ C)$ $\Delta V = 147.303375 mm^3$

The increase in volume is $\boxed{147.30 mm^3}$.

7.2.2 The volume at $88^\circ \text{C}$

Step 1: Calculate the final volume ($V_f$). $V_f = V_0 + \Delta V$ $V_f = 86625 mm^3 + 147.303375 mm^3$ $V_f = 86772.303375 mm^3$

The volume at $88^\circ \text{C}$ is $\boxed{86772.30 mm^3}$.