Chapter 12.2, Problem 27PP

To determine

The number of times the spoon is to be stirred to cause a given temperature change.

Answer to Problem 27PP

The number of times the spoon is to be stirred is $\text{2}\text{.6}\times {\text{10}}^4$

Explanation of Solution

Given:

A spoon is stirred in cup of tea with mass 0.15-kg which increases its temperature by 2⁰C

Formula Used:

According to First Law of thermodynamics, the change in the thermal energy of an object is,

  $\Delta U=Q-W$ ……. (1)

Where,

Q is the heat supplied to the object

W is the work done by the object.

The thermal energy is given by,

  $\Delta U=mC\Delta T$

Here, m represents the mass of the object,C represents specific heat of the object and $\Delta T$ represents change in temperature

Calculation:

Here, work is done on the object which is the cup of tea. The work done by the spoon is given by,

  $W_c=-W_s$

Also, there is no external source of heat energy. Thus, $Q=0$ .

Now, the thermal energy for the tea of mass $m=0.15-\text{kg}$ , specific heat $C=4180\text{ J/kg}\text{.K}$ and with a change in temperature $\Delta T=2°\text{C}$ is obtained as,

  $\begin{array}{c}\Delta U=0.15\times 4187\left(2\right)\\ =1.256\times {10}^3\text{ J}\\ \text{=1}\text{.3}\times {10}^3\text{ J}\end{array}$

Thus, based on (1), the thermal energy is given by,

  $\begin{array}{c}\Delta U=-W_b\\ =-\left(-W_s\right)\\ =W_s\end{array}$

Now, the work done by the spoon for one stir is 0.050 J. If the number of stirs is n, the above equation can be rewritten as,

  $\begin{array}{c}\Delta U=n{W}_{s1}\\ 1.3\times {10}^3=n\times 0.050\\ n=\frac{1.3\times {10}^3}{0.050}\\ =2.6\times {10}^4\end{array}$

Conclusion:

Therefore, the number of stirs needed for the desired temperature change in tea is $2.6\times {10}^4$ .