Calorimetry
3.0 Specific heat capacity and molar heat capacity
3.0 Specific heat capacity and molar heat capacity
If a $Q$ amount of heat is supplied to a body, the temperature of the body increases.
The amount of heat absorbed by the body depends on,
- the mass of the body
- change in temperature
- the material of the body
- and the surroundings
The expression can be written as $$Q = ms\Delta T$$ where
$Q$ is the heat supplied,
$\Delta T$ is the change in temperature,
$m$ is the mass of the body,
$s$ is a constant of the material under the given surrounding conditions.
The constant $s$ is called the specific heat capacity of the substance.
It can also be defined as the amount of heat required for a unit mass of the substance to raise its temperature by ${1^o}C$. $$s = \frac{1}{m}\frac{{\Delta Q}}{{\Delta T}}$$ The specific heat of a substance is proportional to the heat required to change its temperature.
The substance with high specific heat requires more heat than a substance with low specific heat to change its temperature.
The SI unit of specific heat is $Joule/kg-K$ or ( $J\,k{g^{ - 1}}\,{K^{ - 1}}$)
The CGS unit is $cal/g{\,^o}C\;\left( {cal\,{g^{ - 1}}{\,^o}{C^{ - 1}}} \right)$.
Question 1. A 10$g$ copper block is heated till its temperature is raised by ${30^o}C$. Find the heat supplied to the block.( specific heat of copper= $0.09\,cal\,{g^{ - 1}}{\,^o}{C^{ - 1}}$ ).
Solution:
Heat supplied = $$\begin{equation} \begin{aligned} Q = ms\Delta T \\ = \left( {60\,g} \right)\left( {0.09\,cal\,{g^{ - 1}}{\,^o}{C^{ - 1}}} \right)\left( {{{20}^o}C\,} \right) = 108\,cal \\\end{aligned} \end{equation} $$
Molar specific heat
If the amount of substance is measured in moles, then$$Q = nC\Delta T$$ where
$C$ is the molar specific heat of the substance,
$n$ is the number of moles.
It can also be defined as the amount of heat needed to raise the temperature of one mole of substance by ${1^o}C$.$$\begin{equation} \begin{aligned} C = \frac{M}{m}\frac{{\Delta Q}}{{\Delta T}} \\ C = M \times s \\\end{aligned} \end{equation} $$The SI unit of molar specific heat is $J\,mo{l^{ - 1}}\,{K^{ - 1}}$.
The molar specific heat of all metals at room temperature is constant.
It is equal to $25\,J\,mo{l^{ - 1}}\,{K^{ - 1}}$.
Question 2. Find the molar heat capacity of the process $P = \frac{a}{T}$ for a monatomic gas, '$a$' being positive constant.
Solution:
We know that $$dQ = dU + dW$$Specific heat $$C = \frac{{dQ}}{{ndT}} = \frac{{dU}}{{ndT}} + \frac{{dU}}{{ndT}}$$ Since, $$\begin{equation} \begin{aligned} dU = n{C_V}dT \\ C = {C_V} + \frac{{dW}}{{ndT}} \\ C = {C_V} + \frac{{PdV}}{{ndT}} \\ PV = RT \\\end{aligned} \end{equation} $$ Thus for this process, $$\begin{equation} \begin{aligned} V = \frac{{nRT}}{P} = \frac{{nR{T^2}}}{a}\quad \\ \frac{{dV}}{{dT}} = \frac{{2nRT}}{a} \\ \\ \therefore \;C = {C_V} + \frac{P}{n}\left( {\frac{{2nRT}}{a}} \right) \\ C = {C_V} + 2R \\ C = \frac{3}{2}R + 2R \\ C = \frac{7}{2}R \\\end{aligned} \end{equation} $$
