Question

A water heater can generate 32000 kJ/h. How much water can it heat from \(12{\rm{\circ C}}\)to \(14{\rm{\circ C}}\)

Short answer

It can heat \(254.82\,\,{\rm{kg}}\) water per hour.

Step-by-step solution

Definition of heat capacity

The heat capacity of a system represents the heat energy required to raise the temperature of the system.The heat capacity of the system is dependent on the mass and temperature of the system. In other words, the mass and temperature of the system are directly proportional to the heat capacity of the system.

The expression for the heat generated by the water heater per hour is

\(Q = mc\left( {{T_{\rm{f}}} - {T_{\rm{i}}}} \right)\).

Here, \(m\)is mass of water, \(c\)is the specific heat capacity of water, \({T_{\rm{f}}}\)is the final temperature of water, and \({T_{\rm{i}}}\) is the initial temperature of water.

Given information

The final temperature of the water is\({T_{\rm{f}}} = 42{\rm{\circ C}}\).

The initial temperature of the water is\({T_{\rm{i}}} = 12{\rm{\circ C}}\).

The heat capacity of water is \(Q = 32000\,{\rm{kJ}}\).

Calculation of the mass of the water per hour

The mass of water can be calculated as shown below:

\(\begin{array}{c}m = \frac{Q}{{c\left( {{T_{\rm{f}}} - {T_{\rm{i}}}} \right)}}\\ = \frac{{\left( {32000\,{\rm{kJ}}} \right)\left( {\frac{{\left( {{{10}3}\,{\rm{J}}} \right)}}{{1\,{\rm{kJ}}}}} \right)}}{{\left( {4816\,\,{{\rm{J}} \mathord{\left/{\vphantom {{\rm{J}} {{\rm{kg}}}}} \right.\\} {{\rm{kg}}}} \cdot {\rm{\circ C}}} \right)\left( {\left( {42\,\circ {\rm{C}}} \right) - \left( {12\,\circ {\rm{C}}} \right)} \right)}}\\ = 254.82\,{\rm{kg}}\end{array}\)

Hence, the mass of water heated per hour is \(254.82\,\,{\rm{kg}}\).