A mass of 2 kg of water at 18°C is poured into a well-insulated container (figure 3.10) whose temperature is 15°C. The temperatures of the water and the container reached equilibrium at 17.4°C. Determine the amount of heat transferred and its conventional direction when we consider as a system: a) the container with the insulation, b) the water, c) the container with the insulation and the water. The specific heat of water is 1 kcal/kgK.

1. Problem Description

A mass of 2 kg of water at 18°C is poured into a well-insulated container (figure 3.10) whose temperature is 15°C. The temperatures of the water and the container reached equilibrium at 17.4°C. Determine the amount of heat transferred and its conventional direction when we consider as a system: a) the container with the insulation, b) the water, c) the container with the insulation and the water. The specific heat of water is 1 kcal/kgK.

2. Solution Steps

a) For the container with the insulation as a system:

The container with insulation is the system. The water is the surroundings. Heat is transferred from the water to the container.

Let $m_w$ be the mass of water, $T_{wi}$ be the initial temperature of water, $T_{cf}$ be the final temperature of the container and water, $c_w$ be the specific heat of water, and $Q_c$ be the heat absorbed by the container.

$m_w = 2 kg$

$T_{wi} = 18°C$

$T_{ci} = 15°C$

$T_{cf} = 17.4°C$

$c_w = 1 kcal/kgK$

Heat lost by water is equal to heat gained by the container. The water temperature decreases from $18°C$ to $17.4°C$.

The heat lost by water is given by:

$Q_w = m_w * c_w * (T_{wi} - T_{cf})$

$Q_w = 2 kg * 1 kcal/kgK * (18°C - 17.4°C) = 2 * 1 * 0.6 = 1.2 kcal$

The heat gained by the container is $Q_c = -Q_w = 1.2 kcal$. The heat transfer is from the water to the container. The heat gained by the container with insulation is equal to $+1.2 kcal$.

b) For the water as a system:

The water is the system. The container with insulation is the surroundings. Heat is transferred from the water to the container.

$Q_w = m_w * c_w * (T_{cf} - T_{wi})$

$Q_w = 2 kg * 1 kcal/kgK * (17.4°C - 18°C) = 2 * 1 * (-0.6) = -1.2 kcal$

The heat lost by the water is $-1.2 kcal$. The heat transfer is from the water to the container.

c) For the container with insulation and the water as a system:

The system is the container with insulation and the water. The surroundings are "nothing". As the container is well-insulated, there is no heat transfer between the system and the surroundings.

$Q = 0$

3. Final Answer

a) The heat gained by the container with insulation is 1.2 kcal. The direction of heat transfer is from water to the container.

b) The heat lost by the water is -1.2 kcal. The direction of heat transfer is from water to the container.

c) The heat transfer is 0 kcal.