Question

A container with $1kg$ of water in it is kept in sunlight, which causes the water to get warmer than the surroundings. The average energy per unit time per unit area received due to the sunlight is $700W{m}^{-2}$ and it is absorbed by the water over an effective area of $0.05m^2$. Assuming that the heat loss from the water to the surroundings is governed by Newton’s law of cooling, the difference $(in<span\; style="font-family:"Cambria\; Math",serif;mso-bidi-font-family:"Cambria\; Math";">℃</span>)$ in the temperature of water and the surroundings after a long time will be _____________.

(Ignore the effect of the container, and take constant for Newton’s law of cooling $=0.001s^{-1}$, the Heat capacity of water $=4200Jk{g}^{-1}{K}^{-1}$)

Answer 1

Step 1. Given data:

$m=1kg$ Weight of water.

$A=0.05m^2$, Effective area of water

$C=4200JK{g}^{-1}{K}^{-1}$, Heat capacity of water

$\begin{array}{rcl}\frac{1}{A}\frac{dQ}{dt}& =& 700W{m}^{-2}\end{array}$, Energy per unit time per unit area

$\begin{array}{rcl}& <span\; style="font-family:"Cambria\; Math",serif;mso-bidi-font-family:"Cambria\; Math";">\Rightarrow </span>& \frac{dQ}{dt}=700\times 0.05\\ <span\; style="font-family:"Cambria\; Math",serif;mso-bidi-font-family:"Cambria\; Math";">\therefore </span>\frac{dQ}{dt}& =& 35W\end{array}$

$\frac{e\sigma A{T}^3}{mC}=0.001s^{-1}$, Constant for Newton's law of cooling

$$\begin{gathered} e\sigma A{T}^3=0.001\times 1\times 4200 \\ e\sigma A{T}^3=4.2 \end{gathered}$$

Step 2. Calculating value of the difference $(in℃)$ in the temperature of water and the surroundings after a long time

$\frac{dQ}{dt}=e\sigma A\left(T^4-{T^4}_o\right)$

For small temperature change,

$\frac{dQ}{dt}=e\sigma A{T}^3∆T$

Putting all values in the above equation,

$\begin{array}{rcl}\frac{dQ}{dt}& =& \left(e\sigma A{T}^3\right)∆T\\ & \Rightarrow & 35=4.2\times ∆T\\ & \Rightarrow & ∆T=\frac{35}{4.2}\\ \therefore ∆T& =& 8.33\end{array}$

Hence, the difference $(in℃)$ in the temperature of water and the surroundings after a long time $\begin{array}{rcl}\mathbf{∆}\mathit{T}\mathbf{}\mathbf{}& \mathbf{=}& \mathbf{}\mathbf{8}\mathbf{.}\mathbf{33}\end{array}$.