Passage:

Solubility of gases in liquids is greatly affected by pressure and temperature. Henry gave the quantitative relationship between pressure and solubility of gas in a solvent. According to Henry’s law, partial pressure of a gas above a liquid is directly proportional to its mole fraction in solution and is expressed as \(P = K_H.x\), where \(K_H\) is Henry’s constant and x is mole fraction of gas. \(K_H\) is a function of nature of gas.

F M Raoult gave a quantitative relationship between partial pressures and mole fractions in the binary solution of volatile liquids. Raoult’s law states that for a binary solution of volatile liquids, the partial pressure of each component in the solution is directly proportional to its mole fraction. Thus for a solution of component \(1\) and \(2\), partial pressure of each component\(P_1 = p_1^0x_1\), where \(p_1^0\) is the vapour pressure of pure component \(1\) at the same temperature. Similarly \(P_2 = p_2^0x_2\)

\(K_H\) value for \(Ar(g)\), \(CO(g)\), \(HCHO(g)\) and \(CH_4(g)\) are \(40.39\), \(1.67\), \(1.83 × 10^{–5}\) and \(0.413\) respectively. Identify the correct order of their increasing solubility in water

\(Ar < CO_2 < CH_4 < HCHO\)

The correct answer is option 3. \(Ar < CO_2 < CH_4 < HCHO\).

Henry's Law describes the relationship between the solubility of a gas in a liquid and the partial pressure of that gas above the liquid. The mathematical representation of Henry's Law is given by:
\[ C = K_H \cdot P \]
where:
- \(C\) is the concentration of the gas in the liquid (solubility),
- \(K_H\) is Henry's Law constant,
- \(P\) is the partial pressure of the gas above the liquid.

The higher the \(K_H\) value, the more soluble the gas is in the liquid.

Given \(K_H\) values for \(Ar(g)\), \(CO_2(g)\), \(HCHO(g)\), and \(CH_4(g)\):
\(K_H\) for \(Ar(g)\): \(40.39\)
\(K_H\) for \(CO_2(g)\): \(1.67\)
\(K_H\) for \(HCHO(g)\): \(1.83 \times 10^{-5}\)
\(K_H\) for \(CH_4(g)\): \(0.413\)

We want to compare these values to determine the correct order of increasing solubility in water.

The higher the \(K_H\) value, the more soluble the gas is. Therefore, the correct order of increasing solubility is from the highest \(K_H\) to the lowest:

\[ Ar < CO_2 < CH_4 < HCHO \]

So, the correct option is (3) \(Ar < CO_2 < CH_4 < HCHO\). This means that argon (\(Ar\)) is the most soluble, followed by carbon monoxide (\(CO_2\)), methane (\(CH_4\)), and formaldehyde (\(HCHO\)).