Determine the final equilibrium temperature, in °C. A system consists of a copper tank (mass = 17.5 kg, initial temperature = 27°C), 4 kg of liquid water (initial temperature = 50°... Determine the final equilibrium temperature, in °C. A system consists of a copper tank (mass = 17.5 kg, initial temperature = 27°C), 4 kg of liquid water (initial temperature = 50°C), and an electrical resistor. The system is insulated, and the resistor transfers 100 kJ of energy to the system. Read more

Steps to Solve

1. Define the system and its components

The system consists of a copper tank, liquid water, and an electrical resistor. The resistor's mass is negligible. We are given:

• Mass of copper, $m_c = 17.5$ kg
• Mass of water, $m_w = 4$ kg
• Initial temperature of copper, $T_{c,i} = 27^\circ$C
• Initial temperature of water, $T_{w,i} = 50^\circ$C
• Energy added by the resistor, $Q = 100$ kJ $= 100,000$ J

2. Determine the specific heat capacities

We need the specific heat capacities of copper and water:

• Specific heat of copper, $c_c = 386 , \text{J/kg}^\circ\text{C}$
• Specific heat of water, $c_w = 4186 , \text{J/kg}^\circ\text{C}$

3. Apply the energy balance equation

Since the system is insulated, the total energy change is due to the electrical resistor. The energy added by the resistor heats both the copper and the water until they reach a final equilibrium temperature, $T_f$. The change in energy can be written as: $$Q = m_c c_c (T_f - T_{c,i}) + m_w c_w (T_f - T_{w,i})$$

4. Solve for the final temperature

Rearrange the equation to solve for $T_f$:

$Q = m_c c_c T_f - m_c c_c T_{c,i} + m_w c_w T_f - m_w c_w T_{w,i}$

$Q + m_c c_c T_{c,i} + m_w c_w T_{w,i} = T_f (m_c c_c + m_w c_w)$

$T_f = \frac{Q + m_c c_c T_{c,i} + m_w c_w T_{w,i}}{m_c c_c + m_w c_w}$

5. Plug in the values and calculate $T_f$ $$T_f = \frac{100000 + (17.5)(386)(27) + (4)(4186)(50)}{(17.5)(386) + (4)(4186)}$$

$$T_f = \frac{100000 + 182955 + 837200}{6755 + 16744}$$

$$T_f = \frac{1120155}{23499}$$

$$T_f \approx 47.67^\circ\text{C}$$