Chapter 21, Problem 62A

What is the specific heat capacity of a 50-gram piece of $100°C$ metal that will change 400 grams of $20°C$ water to $22°C$?

To determine

The required specific heat capacity of the piece of metal.

Answer to Problem 62A

The required specific heat capacity of the piece of metal is $0.205\text{cal/g}\cdot °\text{C}$ .

Explanation of Solution

Given info:

The mass of the metal $m_{\text{m}}=50\text{g}$ .

The initial temperature of the metal is $T_{\text{1}}=100°\text{C}$ .

The mass of water $m_{\text{w}}=400\text{g}$ .

The initial temperature of the water is $T_2=20°\text{C}$ .

The equilibrium temperature of mixture of hot and cold water is $T=22°\text{C}$ .

Formula used:

The certain amount of heat energy is supplied to increase the temperature of the substance.

The expression for the required amount of heat energy $\left(Q\right)$ depends on the initial temperature $\left(T_{\text{i}}\right)$ of substance, mass $\left(m\right)$ of substance, and specific heat $\left(c\right)$ of substance is,

  $Q=mc\left(T_f-T_i\right)$

Here, $T_f$ is the final temperature of the substance.

The expression for the equal amount of heat lost by hot water to the amount of heat gain by cold water from the equilibrium condition is,

  $\begin{array}{c}{Q}_{\text{lost}}=Q_{\text{gain}}\\ m_{\text{m}}{c}_{\text{m}}\left(T_1-T\right)=m_{\text{w}}{c}_{\text{w}}\left(T-T_2\right)\end{array}$

Here, $c_{\text{m}}$ is the specific heat of metal and $c_{\text{w}}$ is the specific heat of water.

Calculation:

Take the specific heat of the water as $1\text{cal/g}\cdot °\text{C}$ .

The required specific heat capacity of piece of the metal is,

  $\begin{array}{l}{m}_{\text{m}}{c}_{\text{m}}\left(T_1-T\right)=m_{\text{w}}{c}_{\text{w}}\left(T-T_2\right)\\ c_{\text{m}}=\frac{\left(400\text{g}\right)\left(1\text{cal/g}\cdot °\text{C}\right)\left(22°\text{C}-20°\text{C}\right)}{\left(50\text{g}\right)\left(100°\text{C}-22°\text{C}\right)}\\ c_{\text{m}}=0.205\text{cal/g}\cdot °\text{C}\end{array}$

Conclusion:

Thus, the required specific heat capacity of piece of the metal is $0.205\text{cal/g}\cdot °\text{C}$ .