Question

The heat capacity of air is much smaller than that of water, and relatively modest amounts of heat are needed to change its temperature. This is one of the reasons why desert region, although very hot during the day, are bitterly cold at night. The heat capacity of air at room temperature and pressure is appoximately 21 J/K*mol. How much energy is required to raise the temperature of a room of dimensions 5.5m x 6.5m x 3.0m by 10 degrees Celsius? If losses are neglected, how long will it take a heater rated at 1.5 kW to achieve that increase given that 1 W = 1 J/s?

Answer 1

Answer:

$Q=9.2x10^5J$

$t=614s=10.2min$

Explanation:

Hello,

In this case, we can compute the energy by using the following formula for air:

$Q=nCp\Delta T$

Whereas the moles of air are computed via the ideal gas equation at room temperature inside the 5.5m x 6.5m x 3.0m-room:

$n=\frac{PV}{RT}\\\\V=5.5m*6.5m*3.0m=107.25m^3*\frac{1000L}{1m^3}=107250L\\ \\n=\frac{1atm*107250L}{0.082\frac{atm*L}{mol*K}*298.15K}\\ \\n=4386.8mol$

Now, we are able to compute heat, by considering that the temperature raise is given in degree Celsius or Kelvins as well:

$Q=4386.8mol*21\frac{K}{mol*K}*10K \\\\Q=9.2x10^5J$

Finally, we compute the time required for the heating by considering the heating rate and the required heat, shown below:

$t=\frac{9.2x10^5J}{1.5\frac{kJ}{s}*\frac{1000J}{1kJ} } \\\\t=614s=10.2min$

Regards.