Question

A solid cube of mass m at a temperature $θ_0$ is heated at a constant rate. It becomes liquid at temperature $θ_1$ and vapour at temperature $θ_2$ . Let $s_1$ and $s_2$ be specific heats in its solid and liquid states respectively. If $L_f$ and $L_v$ are latent heats of fusion and vaporisation respectively, then the minimum heat energy supplied to the cube until it vaporises is

• A. $ms_1 (θ_1 - θ_0) + ms_2 (θ_2 - θ_1)
• B. $mL_f + Ms_2(θ_2 - θ_1) + mL_v$
• C. $ms_1 (θ_1 - θ_0) +mL_f + ms_2 (θ_2 - θ_1) + mL_f $
• D. $ms_1 (θ_1 - θ_0) +mL_f + ms_2 (θ_2 - θ_0) + mL_f$

Answer 1

The Correct Option is C

Given:
Mass of the cube \( m \)
Initial temperature of the cube \( \theta_0 \)
Final temperature to become liquid \( \theta_1 \)
Final temperature to become vapour \( \theta_2 \)
Specific heat in the solid state \( s_1 \)
Specific heat in the liquid state \( s_2 \)
Latent heat of fusion \( L_f \)
Latent heat of vaporisation \( L_v \) The minimum heat energy required to heat the cube from its initial state to the point where it turns into vapour consists of the following stages: 1. Heating the solid cube from temperature \( \theta_0 \) to \( \theta_1 \): The heat required for this stage is: \[ Q_1 = m s_1 (\theta_1 - \theta_0) \] 2. Melting the cube from solid to liquid at temperature \( \theta_1 \): The heat required for this stage is the latent heat of fusion: \[ Q_2 = m L_f \] 3. Heating the liquid from temperature \( \theta_1 \) to \( \theta_2 \): The heat required for this stage is: \[ Q_3 = m s_2 (\theta_2 - \theta_1) \] 4. Vaporising the liquid at temperature \( \theta_2 \): The heat required for this stage is the latent heat of vaporisation: \[ Q_4 = m L_v \] Total heat energy required: The total heat energy supplied to the cube is the sum of all these stages: \[ Q_{\text{total}} = m s_1 (\theta_1 - \theta_0) + m L_f + m s_2 (\theta_2 - \theta_1) + m L_v \] Thus, the minimum heat energy supplied to the cube until it vaporises is: \[ Q_{\text{total}} = m s_1 (\theta_1 - \theta_0) + m L_f + m s_2 (\theta_2 - \theta_1) + m L_v \] Therefore, the correct answer is (C): \( m s_1 (\theta_1 - \theta_0) + m L_f + m s_2 (\theta_2 - \theta_1) + m L_v \).

Answer 2

To calculate the minimum heat energy required to heat the solid cube until it vaporises, we need to account for the following stages:
1. Heating the solid cube from \( \theta_0 \) to \( \theta_1 \): The heat required to raise the temperature of the solid from \( \theta_0 \) to \( \theta_1 \) is given by: \[ Q_1 = m s_1 (\theta_1 - \theta_0) \] where \( s_1 \) is the specific heat of the solid and \( m \) is the mass of the cube. 
2. Fusion of the solid to liquid at \( \theta_1 \): The heat required to melt the solid cube at temperature \( \theta_1 \) is: \[ Q_2 = m L_f \] where \( L_f \) is the latent heat of fusion.
3. Heating the liquid from \( \theta_1 \) to \( \theta_2 \): The heat required to raise the temperature of the liquid from \( \theta_1 \) to \( \theta_2 \) is: \[ Q_3 = m s_2 (\theta_2 - \theta_1) \] where \( s_2 \) is the specific heat of the liquid.
4. Vaporisation of the liquid at \( \theta_2 \): The heat required to vaporise the liquid at temperature \( \theta_2 \) is: \[ Q_4 = m L_v \] where \( L_v \) is the latent heat of vaporisation. The total minimum heat energy required is the sum of all these heats: \[ Q_{\text{total}} = Q_1 + Q_2 + Q_3 + Q_4 = m s_1 (\theta_1 - \theta_0) + m L_f + m s_2 (\theta_2 - \theta_1) + m L_v \]

Thus, the minimum heat energy required is given by option (C).