Chapter 12, Problem 46A

To determine

The final temperature of the mixture.

Answer to Problem 46A

The final temperature of the mixture is $62.8°\text{C}$.

Explanation of Solution

Given:

A $4.0\times {10}^2 \text{g}$ of water at a temperature of $22.0°\text{C}$ is mixed with $6.0\times {10}^2 \text{g}$ of water at a temperature of $90°\text{C}$ .There is no loss of energy to container or sorroundings.

Formula Used:

When a hot object at a temperature $T_A$ (in Kelvin) is immersed in cooler water at initial temperature $T_B$ (Kelvin) , the temperature of water increases and reaches an equilibrium value. This is due to the exchange of heat energy between the object and water. It is also being assumed that there is no loss of heat to sorroundings or the container.

Now, if the mass of the object and water is $m_A$ and $m_B$ respectively, with specific heat $C_A$ and $C_B$ , the equilibrium temperature is given by the expression,

  $T_f=\frac{m_A{C}_A{T}_A+m_B{C}_B{T}_B}{m_A{C}_A+m_B{C}_B}$ ..... (1)

Calculation:

Here, the two components of the mixture are water. The hot water which has a mass of $m_A\text{=6}.0\times \text{1}{0}^{\text{2}}–\text{g}$ is kept at a temperature 90.0⁰C. The cold water has a mass of $m_B\text{=4}.00\times \text{1}{0}^{\text{2}}–\text{g}$ and is at a temperature 22.0⁰C. The specific heat is $C_A=C_B=4180\text{ J/kg}\text{.K}$ .

The temperature of hot water in Kelvin can be calculated as,

  $\begin{array}{c}{T}_A=90+273\\ =363\text{ K}\end{array}$

Similarly, the temperature of cold water in Kelvin is,

  $\begin{array}{c}{T}_A=22+273\\ =295\text{ K}\end{array}$

Substituting the given parameters in equation (1), equilibrium temperature is obtained as,

  $\begin{array}{c}{T}_f=\frac{6\times {10}^2\times 4180\times 363+4.0\times {10}^2\times 4180\times 295}{6\times {10}^2\times 4180+4.0\times {10}^2\times 4180}\\ =335.8\text{ K}\end{array}$

Thus, the equilibrium temperature in degree Celsius is,

  $\begin{array}{c}{T}_f=335.8-273\\ =62.8°\text{C}\end{array}$

Conclusion:

Thus, the equilibrium temperature of mixture is $62.8°\text{C}$ .