Question

The specific heat capacity of water is $4200 Jkg^{-1}{}^{\circ}C^{-1}$. Calculate the heat capacity of $10\; kg$ of water per ${}^{\circ}C$

Answer 1

The easiest way to go about this would be to infer the relation by putting together the definitions of heat capacity and specific heat capacity. Heuristically, we know that the heat capacity is dependent on the mass of the body while specific heat is a characteristic of the material of the body irrespective of its mass, and is defined per unit mass. Putting these two together mathematically should yield the necessary relation and a subsequent solution.

Formula Used:
The heat capacity of a body $C_{H} = mC_{S}$, where $C_S$ is the specific heat of the material of the body and m is the mass of the body.

Complete step-by-step solution:
Let us begin by establishing a distinction between the terms heat capacity and specific heat capacity.
Heat capacity, also known as thermal capacity is a measurable physical quantity that characterizes the amount of heat required to raise the temperature of the substance by $1^{\circ}C$, and is given as the ratio of the amount of heat energy Q transferred to a substance to the resulting increase in the temperature ($\Delta T$) of the substance, i.e.,
$C_{H} = \dfrac{Q}{\Delta T}$ and is usually measured in $J^{\circ}C^{-1}$
Now, the specific heat capacity, also known as specific heat is defined as the amount of heat that is required to raise the temperature of a unit mass by $1^{\circ}C$ and is given by:
$C_{S} = \dfrac{Q}{m\Delta T}$ and is usually measured in $Jkg^{-1} {}^{\circ}C$
As its name suggests specific heat is specific to that substance.
Now, from the above formulation we can see that the relationship between heat capacity and the specific heat is as follows:
$C_{H} = mC_{S}$ which makes sense because the overall capacity of a body to hold heat is ultimately dependent on the mass of the body times the amount of heat it can hold in every unit mass.
Therefore, substituting the values from the question, we get:
$C_{H} = 10 \times 4200 = 42000 = 42kJ^{\circ}C^{-1}$
Therefore, the heat capacity of 10 kgs of water is $42kJ^{\circ}C^{-1}$.

Note: Thermodynamically speaking, heat capacity is an extensive variable, which means that the quantity of heat a substance can hold depends on the quantity of matter that constitutes the substance. Thus, heat capacity scales with the size of the system. However, specific heat is an intensive variable, since it describes the amount of heat that is needed to be supplied to a unit of mass, and hence is independent of the total mass of the substance.