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# Gas Laws

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 Sub Topics The Gas laws were proposed very early at the end of the 18th century, when scientists started realizing the relationships between the pressure, volume and temperature of any sample of gas. It came out of the basic logic that the state of a gas is a function of the pressure, volume and temperature. If one quantity is varied keeping the other quantities constant, we get gas laws. There are basically five gas laws namely,Boyle's LawCharles LawGay Lussac LawAvogadro LawHenrys LawLets study about them.

## Boyle's Law

Its named came after Robert Boyle that gives the relation between the volume and the pressure of a given mass of a gas at constant temperature which states that :
At constant temperature, the pressure of an ideal gas varies inversely to its volume. At constant temperature T, if P is the pressure of a certain mass of a gas and V is its volume at a temperature T K then
P $\propto$ $\frac{1}{V}$

If the pressure is P1 at volume V1 and pressure is P2 at volume V2 then,
P1V1 = P2V2
$\therefore$ PV = constant.

## Charles' law

Its named came after Jacques Charles. He carried on series of experiment to test the Boyle's law, to determine the effect of temperature changes that have on gases. He discovered the relationship between temperature and volume at constant temperature which states that:
At constant pressure, the volume of an ideal gas varies directly to the absolute temperature.
V $\propto$ T

If the volume is V1 at temperature T1 and volume is V2 at temperature T2 then,
$\frac{V_1}{T_1}$ = $\frac{V_2}{T_2}$

$\therefore$ $\frac{V}{T}$ = constant.

## Gay-Lussac's Law

In 1787, Jacques Charles studied the effect of change of temperature on the volume of a given mass of a gas at constant pressure, but he did not publish his results, Joseph Louis Gay-Lussac discovered in 1802 that the volume of a gas increased linearly with increase of temperature if the pressure is kept constant . Hence, this laws is also called Gay Lussac Law.
Mathematically, it is given as pressure P is proportional to temperature T given as,
P $\propto$ T

$\frac{P_1}{T_1}$
= $\frac{P_2}{T_2}$

$\therefore$ $\frac{P}{T}$ = Constant.

In 1811, Amedeo Avogadro proposed the law, that gives a relationship between the volume of a gas and the number of molecules in it at a given temperature and pressure which states that:
Equal volumes of gases at the same temperature and pressure contain same number of molecules regardless of molecules. It is stated as,
n1 = n2

where, n1 is the number of molecules in one gas and n2 is the number of molecules in another gas

Hence, the equal volumes V of all gases if measured under the same temperature T and pressure P conditions will have equal number of molecules n.

Mathematically,
V $\propto$ n

$\frac{V_1}{n_1}$
= $\frac{V_2}{n_2}$

$\frac{V}{n}$
= constant.

## Combined Gas Law

According to Boyle's law, at constant temperature
PV = constant
or
P1V1 = P2V2

According to Charles law, at constant pressure
V $\propto$ T

If the volume is V1 at temperature T1 and volume is V2 at temperature T2 then
$\frac{V_1}{T_1}$ = $\frac{V_2}{T_2}$
$\therefore$ $\frac{V}{T}$ = constant.

According to Gay-Lussac's law, also known as pressure law
P $\propto$ T
$\frac{P_1}{T_1}$ = $\frac{P_2}{T_2}$
$\therefore$ $\frac{P}{T}$ = Constant.

Now, according to Avogadro's law
$\frac{V}{n}$ = constant

By introducing the Avogadro's law in the combined gas law equation, we get
$\frac{PV}{nT}$ = constant
The constant is represented by R
So by rearranging, we obtain the expression for the ideal gas law
PV = nRT

## Gas law Problems

Lets see some examples on gas law problems:

### Solved Examples

Question 1: An air bubble is released at the bottom of a lake where the temperature is $4^{0}$C and the pressure is 3.40 atm. If the bubble was 10.0 mL to start, what will it’s volume be at the surface, where the water temperature is $12^{0}$C and the pressure is $10^{3}$ kPa.
Solution:

Given that,
P1 = 3.40 atm
P2 = 103 kPa = 1.0165 atm
T1 = 40 C = 277 K
T2 = 120 C = 285 K
V1 = 10 mL = 0.01 L
V2 = ?

Using $\frac{P_1 V_1}{T_1}$ = $\frac{P_2 V_2}{T_2}$
$\frac{3.4 atm \times 0.01 L}{277 K}$ = $\frac{1.0165 atm \times V_2}{285 K}$
V2 = 0.0344 L = 34.4 mL.

Question 2: What is the temperature of 0.70 moles of a gas that occupies 0.47 L at a pressure of 150 kPa?
Solution:

According to Ideal Gas Law,
PV = nRT

By rearranging,
T = $\frac{PV}{nR}$
T = $\frac{150 kPa \times 0.47 L}{0.7 mol \times 8.3}$
T = 12 K.