Question

The molar heat capacity of water at constant pressure, ${{\text{C}}_{\text{p}}}$ is ${\text{7}}{\text{.5}}\,{\text{J}}\,\,{{\text{K}}^{ - 1}}\,{\text{mo}}{{\text{l}}^{ - 1}}$. when $10$ kJ of heat is supplied to$1$ kg water which is free to expand, the increase in temperature of water is:
A. ${\text{2}}{\text{.4}}\,{\text{K}}$
B.${\text{4}}{\text{.8K}}$
C.${\text{3}}{\text{.2K}}$
D.${\text{10K}}$

Answer 1

We will determine the mole of water by using mole formula. Then by using a heat formula we can determine the temperature increases. The heat required for the increase in temperature is determined as the product of mass and heat capacity of that compound and temperature difference.

Complete solution:
The heat of fusion represents the heat taken by a solid to convert into liquid. The heat which will be taken by solid, here, ice will be provided by some source, here, water. So, the solid will gain heat and the water will lose the heat.
The formula used to determine the heat change is as follows:
${\text{q}}\,{\text{ = }}\,{\text{m}}{{\text{C}}_{\text{p}}}{\delta T}$
Where,
${\text{q}}$is the heat.
${\text{m}}$is the mass of the substance
${{\text{C}}_{\text{p}}}$is the heat capacity of the substance
${\delta T}$is the change in temperature
Heat capacity of water at constant pressure, ${{\text{C}}_{\text{p}}}$ is ${\text{7}}{\text{.5}}\,{\text{J}}\,\,{{\text{K}}^{ - 1}}\,{\text{mo}}{{\text{l}}^{ - 1}}$. when $10$ kJ of heat is supplied to$1$ kg water which is free to expand, the increase in temperature of water is:
First, we will convert the heat supplied from kJ to joule as follows:
$1000$ kJ = $1$J
$10$ kJ = $0.010$J
We will convert the amount of water from kg to gram as follows:
$1$ kg = $1000$g
Now, we will determine the mole of water by using the mole formula as follows:
${\text{mole}}\,{\text{ = }}\,\dfrac{{{\text{mass}}}}{{{\text{molar}}\,{\text{mass}}}}$
Molar mass of water is $18$ g/mol.
On substituting $1000$g for mass and $18$ g/mole for molar mass of water,
${\text{mole}}\,{\text{ = }}\,\dfrac{{1000}}{{{\text{18}}}}$
${\text{mole}}\,{\text{ = }}\,{\text{55}}{\text{.55}}\,{\text{mol}}$
So, the amount of water is ${\text{55}}{\text{.55}}$mol.
Now we will determine the increase in temperature as follows:
${\text{q}}\,{\text{ = }}\,{\text{m}}{{\text{C}}_{\text{p}}}{\delta T}$
On substituting ${\text{55}}{\text{.55}}$for the mass of water, ${\text{7}}{\text{.5}}\,{\text{J}}\,\,{{\text{K}}^{ - 1}}\,{\text{mo}}{{\text{l}}^{ - 1}}$for the heat capacity of water and $0.010$J for heat supplied.
$0.010{\text{J}}\, = \,{\text{55}}{\text{.55}}\,{\text{mol}} \times {\text{7}}{\text{.5}}\,{\text{J}}\,\,{{\text{K}}^{ - 1}}\,{\text{mo}}{{\text{l}}^{ - 1}} \times {\delta T}$
${\delta T}\, = \,\dfrac{{0.010{\text{J}}}}{{{\text{55}}{\text{.55}}\,{\text{mol}} \times {\text{7}}{\text{.5}}\,{\text{J}}\,\,{{\text{K}}^{ - 1}}\,{\text{mo}}{{\text{l}}^{ - 1}}}}$
${\delta T}\, = \,\dfrac{{0.010}}{{416.6\,{{\text{K}}^{ - 1}}\,}}$
${\delta T}\, = \,2.4\,{\text{K}}$
So, the increase in temperature of water is $2.4$ K.

Therefore, option (A) is correct.

Note: The heat required to increase the temperature of a substance by one degree Celsius is known as heat capacity. The heat required to increase the temperature of one gram substance by one degree Celsius is known as specific heat capacity. The molar heat capacity of water at constant pressure, ${{\text{C}}_{\text{p}}}$ is ${\text{7}}{\text{.5}}\,{\text{J}}\,\,{{\text{K}}^{ - 1}}\,{\text{mo}}{{\text{l}}^{ - 1}}$ it means the energy required to increase the temperature of one mole of water by one kelvin is ${\text{7}}{\text{.5J}}$ . The unit of mass of compound moles are very important.