Specific Heat Of A Gas

Definition of Specific heat

Specific heat - The specific heat is the amount of heat per unit mass required to raise the temperature by one Kelvin.

Now for gases, we have several types of specific heat, but here we will discuss basically two types of specific heat

Specific heat at constant volume(c_v)

Specific heat at constant volume is defined as the quantity of heat required to raise the temperature of a unit mass of gas through 1°C or 1 Kelvin at constant volume.

$\text { It is given as - } c_v=\frac{(\Delta Q)_V}{m \Delta T}$

If 1 mole of gas is placed at the place of unit mass is considered, then this specific heat of gas is called molar specific heat at constant volume and is represented by Cv (Here C is capital)

So, for molar-specific heat -

$ C_V=M c_V=\frac{M(\Delta Q)_V}{m \Delta T}=\frac{1}{\mu} \frac{(\Delta Q)_V}{\Delta T} \quad\left[\text { As } \mu=\frac{m}{M}\right]$

Specific heat at constant pressure

Specific heat at constant pressure (c_p) is defined as the quantity of heat required to raise the temperature of a unit mass of gas through 1°C or 1 Kelvin at constant pressure.

It is qiven as -
$
c_p=\frac{(\Delta Q)_p}{m \Delta T}
$

If 1 mole of gas is placed at the place of unit mass is considered, then this specific heat of gas is called molar specific heat at constant pressure and is represented by C_p (Here C is capital)

So, for molar-specific heat at constant pressure

$ C_p=M c_p=\frac{M(\Delta Q)_p}{m \Delta T}=\frac{1}{\mu} \frac{(\Delta Q)_p}{\Delta T} \quad\left[\text { As } \mu=\frac{m}{M}\right]$

We can understand better through video.

Solved Examples Based on the Specific Heat of a Gas

Example 1: The specific heats $C_p$ and $C_v$ of a gas of diatomic molecules A are given ( in units of $\mathrm{J} \mathrm{mol}^{-1}$ ) by 29 and 22 respectively. Another gas of diatomic molecule B has the corresponding values of 30 and 21 respectively. If they are treated as ideal gases then :

1) A is rigid but B has a vibrational mode.

2) A has a vibrational mode but B has none.

3) A has one vibrational mode and B has two.

4) Both A and B have a vibrational mode each.

Solution:

Specific heat capacity at constant pressure -

$
\begin{aligned}
& C_p=C_v+R \\
& =\left(\frac{f}{2}+1\right) R
\end{aligned}
$
wherein
$\mathrm{f}=$ degree of freedom
$\mathrm{R}=$ Universal gas constant
$
\frac{C_p}{C_v}=1+\frac{2}{f}
$

For gas A,

$\begin{aligned}
& \frac{29}{22}=1+\frac{2}{f} \Rightarrow \frac{2}{f}=\frac{7}{22} \\
& \Rightarrow f=\frac{44}{7} \simeq 6
\end{aligned}$

3 translations, 2 rotations, Remaining vibrational mode

For gas B ,

$\begin{aligned}
& \frac{C_p}{C_v}=\frac{30}{21}=1+\frac{2}{f} \Rightarrow \frac{2}{f}=\frac{9}{21} \\
& \Rightarrow f=\frac{42}{9} \simeq 5
\end{aligned}$

3 translations, 2 rotations, no vibrational mode

Hence, the answer is the option (2)

Example 2: Consider two ideal diatomic gases $\mathrm{A}$ and $\mathrm{B}$ at the same temperature T. Molecules of the gas $A$ are rigid and have a mass of $m$. Molecules of gas B have an additional vibrational mode and have a mass $m$ $\frac{m}{4}$. The ratio of the specific heats $\left(C_V^A\right.$ and $\left.C_V^B\right)$ of gas $\mathrm{A}$ and $\mathrm{B}$ respectively is:

1) 5:9

2) 7:9

3) 3:5

4) 5:7

Solution:

Molar heat capacity of $\mathrm{A}$ at constant volume $=\frac{5 R}{2}$
Molar heat capacity of $\mathrm{B}$ at constant volume $=\frac{7 R}{2}$
Dividing both
$
\frac{\left(C_V\right)_A}{\left.C_V\right)_B}=\frac{5}{7}
$

Hence, the answer is the option (4).

Example 3: Two moles of helium gas are mixed with three moles of hydrogen molecules (taken to be rigid). What is the molar specific heat (in J/mol K) of the mixture at constant volume? (R=8.3 J/mol K)

1) 15.7

2) 17.4

3) 19.7

4) 21.6

Solution:

$\begin{aligned}
& C_{v \text { mix }}=\frac{n_1 C_{v_1}+n_2 C_{v_2}}{n_1+n_2} \\
& =\frac{2 \times \frac{3}{2} R+3 \times \frac{5}{2} R}{2+3} \\
& =\frac{3 R+15 \frac{R}{2}}{5} \\
& =\left(\frac{21}{10}\right) R=\frac{21}{10} \times 8.314 \\
& =17.4 \mathrm{~J} / \mathrm{molK}
\end{aligned}$

Hence the answer is the option (2).

Example 4: A geyser heats water flowing at a rate of $2.0 \mathrm{~kg}$ per minute from $30^{\circ} \mathrm{C}$ to $70^{\circ} \mathrm{C}$. If the geyser operates on a gas burner, the rate of combustion of fuel will be $\qquad$ $g \min ^{-1}$
[Heat of combustion $=8 \times 10^3 \mathrm{Jg}^{-1}$,
Specific heat of water $=4.2 \mathrm{Jg}^{-1}{ }^0 \mathrm{C}^{-1} \mathrm{]}$

1) 42

2) 43

3) 44

4) 45

Solution:

The energy required for heating water in one minute $(\Delta t=1 \mathrm{~min})$
$
\begin{aligned}
\frac{\mathrm{E}}{\Delta \mathrm{t}} & =\frac{\mathrm{mc} \Delta \mathrm{T}}{\Delta \mathrm{t}} \\
& =\frac{2000 \times 4.2 \times 40}{1 \mathrm{~min}} \\
\frac{\mathrm{E}}{\Delta \mathrm{t}} & =336 \times 10^3\left(\frac{\mathrm{J}}{\mathrm{min}}\right)
\end{aligned}
$

The rate of combustion of fuel is let's say $R$
$
\begin{aligned}
\therefore \mathrm{R} & =\frac{(\mathrm{E} / \Delta \mathrm{t})}{\text { Heat of combustion }} \\
& =\frac{336 \times 10^3 \mathrm{~J} / \mathrm{min}}{8 \times 10^3 \mathrm{Jg}^{-1}} \\
\mathrm{R}= & 42\left(\frac{\mathrm{g}}{\mathrm{min}}\right)
\end{aligned}
$

Hence, the answer is option (1).

Example 5 : A cylinder of a fixed capacity of 44.8 liters contains helium gas at standard temperature and pressure. The amount of heat needed to raise the temperature of gas in the cylinder by $20.0^{\circ} \mathrm{C}$ will be :
(Given gas constant $\mathrm{R}=8.3 \mathrm{JK}^{-1}-\mathrm{mol}^{-1}$ )

1) $249 \mathrm{~J}$
2) $415 \mathrm{~J}$
3) $498 \mathrm{~J}$
4) $830 \mathrm{~J}$

Solution:

Since the Isochoric process,

$
\begin{array}{r}
\Delta \mathrm{Q}=\mathrm{nC}_{\mathrm{V}} \Delta \mathrm{T} \\
C_V=\frac{f}{2} R
\end{array}
$

Helium is a monoatomic gas.
Degree of freedom $(f)=3$
44.8 litre at STP $=2$ mole
$
\begin{aligned}
\Delta \mathrm{Q}= & 2 \times \frac{3 \mathrm{R}}{2} \times 20 \\
& =498 \mathrm{~J}
\end{aligned}
$

Hence, the answer is the option (3).

Summary

The specific heat of gas refers to the amount of heat which is needed in order to raise the temperature of a specific amount of gas by one degree; this property is different for gases heated under constant volume (cv) or constant pressure (cp). For gases to absorb and release energy during these processes, they must have certain heat capacities.

Frequently Asked Questions (FAQs):

Q 1: Why is Cp greater than Cv for an ideal gas?

Ans: Cp is greater than Cv because, at constant pressure, the gas does work to expand against the external pressure as it is heated, in addition to increasing its internal energy. At constant volume, no work is done, so all the heat added increases the internal energy of the gas, resulting in a lower specific heat value.

Q 2: What is the specific heat capacity?

Ans: Specific heat capacity is the amount of heat energy that is needed by a substance of one kilogram to raise its temperature by 1 degree Celsius.

Q 3: What are the applications of specific heat capacity in daily life?

Ans: The application of specific heat in daily life includes substances that show a small value for specific heat capacities. These are considered useful substances such as frying pans, pots, and more. This is because of the fast heating of materials, even when a small amount of heat is supplied to them.

Q 4: What is the molar specific heat capacity of a gas at constant volume?

Ans: The molar-specific heat capacity of a gas at constant volume can be described as the amount of energy in the form of heat that is required to raise the temperature of 1 mol of the gas by a factor of one-degree Celsius at constant volume