2.0 Gas laws

  Gaseous State

Some measurable parameters of a gas like pressure, temperature, volume and amount of gas are interrelated and can be expressed in terms of each other by means of different gas laws.

1. Boyle's law: This law states that "At constant temperature, the pressure of a fixed amount of gas varies inversely with its volume". Mathematically, $$P \propto 1/V \ ( with \ n \ and\ T \ constant)$$ $$ \Rightarrow P = {k_1} \times 1/V \ \ \ where \ ,\ k_1 \ is \ constant$$ Graph obtained between $P$ and $1/V$ is called isotherm because across this curve temperature is constant. Suppose two container having volume $V_1$ and $ V_2$ have pressure $ P_1$ and $P_2$ respectively. These two containers are maintained at same temperature and contain same amount of gas. Therefore, from boyle's law, $${P_1}{V_1} = {P_2}{V_2}$$ For a given mass ($m$) of gas, Boyle's law can also be expressed in terms of density ($\rho $).$$\begin{equation} \begin{aligned} PV/m = {k_1}/m \\ P/\rho = k \\\end{aligned} \end{equation} $$

We can write the boyle's law in another form by taking ln (log) i.e., $$PV=constant(c)$$ Take ln both sides, we get

$$\begin{equation} \begin{aligned} \ln (PV) = \ln (c) \\ \ln (P) + \ln (V) = \ln (c) \\ \ln (P) = - \ln (V) + \ln (c) \\\end{aligned} \end{equation} $$

To plot the graph between ln($P$) and ln($V$), compare the equation with $y=mx+c$, we can draw the straight line with negative slope $-1$ as shown in figure.

2. Charle's law: This law states that "At constant pressure, the temperature( in absolute scale) of a fixed amount of gas is directly proportional to its volume". Mathematically, $$T \propto V$$ $$T = kV \ ( \ where\ ,\ k\ is \ consant\ )$$ Curve obtained on plotting $T$ and $V$ is called isobar as pressure is constant across this curve. During an experiment, it was observed that on increasing temperature of a substance by ${1^ \circ }C$ its volume increases by $1/273.15$ times of its volume at ${0^ \circ }C$ . Let us assume that volume of a gas at ${0^ \circ }C$ and at ${t^ \circ }C$ are $V_0$ and $V_t$ respectively, then $$\begin{equation} \begin{aligned} {V_t} = {V_o} + t \times \frac{{{V_o}}}{{273.15}} \\ {V_t} = {V_o}(\frac{{t + 273.15}}{{273.15}}) \\\end{aligned} \end{equation} $$ At this stage, absolute scale was defined .$$T = {t^ \circ }C + 273.15$$ Now, above expression becomes$$\frac{{{V_t}}}{{{V_0}}} = \frac{{{T_t}}}{{{T_0}}}$$In other form, $$T = kV$$

3. Gay Lussac’s Law: This law states that "At constant volume, the temperature (in absolute scale) of a fixed amount of gas is directly proportional to its pressure". Mathematically, $$T \propto P$$ $$T = kP \ ( \ where\ ,\ k\ is \ consant\ )$$ Curve obtained on plotting $P$ and $T$ is called isochor as volume is constant across this curve. Suppose two container having same volume, temperature of the containers be $T_1$ and $ T_2$ and gases exert pressure $ P_1$ and $P_2$ respectively. These two containers contain same amount of gas. Therefore, from Gay Lussac’s Law, $${P_1}/{P_2} = {T_1}/{T_2}$$

4. Avogadro Law: This law states that 'Equal volumes of all gases under the same conditions of temperature and pressure contain equal number of molecules'. Mathematically, $$V \propto n$$Molar volume of a gas at NTP is $22.4$ litres while at STP is $22.7$ litres.

Question 1. Arrange the following:

(a) temperature in descending order. (b) pressure in descending order.

Solution: (a) $T_1$>$T_2$>$T_3$

Draw a horizontal line. This line will act as isobar and this curve will follow charle's law. Now, vertical lines are dropped from the point of intersection of this horizontal line with the other three lines. Suppose intersection of these lines with volume axis represent volume $V_1$, $V_2$ and $V_3$ respectively. The variables $V_i$ and $T_i$ now related according to charle's law. From graph, it is clear that $V_1$>$V_2$>$V_3$. Therefore by charle's law $T_1$>$T_2$>$T_3$.

(b) $P_1$>$P_2$>$P_3$

Draw a horizontal line. This line will act as isotherm and this curve will follow boyle's law. Now, vertical lines are dropped from the point of intersection of this horizontal line with the other three lines. Suppose intersection of these lines with volume axis represent volume $V_1$, $V_2$ and $V_3$ respectively. The variables $V_i$ and $P_i$ now related according to boyle's law. From graph, it is clear that $V_3$>$V_2$>$V_1$. Therefore, by boyle's law $P_1$>$P_2$>$P_3$.