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The Henry's law constant for the solubility of $\ce{N2}$ gas in water at $298~\mathrm{K}$ is $10^5~\mathrm{atm}$. The mole fraction of $\ce{N2}$ in air is $0.8$. The amount of substance of $\ce{N2}$ for air dissolved in 10 moles of water at $298~\mathrm{K}$ and $5~\mathrm{atm}$ pressure is ___.

I think the mole fraction of nitrogen will be calculated using the formula $$x(\ce{N2})=P/K_h.$$ But the answer is wrong. In the solution for the question, the mole fraction $x(\ce{N2})$ is multiplied further by $0.8$ which seems to the mole fraction of nitrogen in air.

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  • $\begingroup$ For all such calculations you have to use the partial pressure of the respective chemical species, not the absolute pressure of the gas phase. Hence that factor ~0.8. $\endgroup$ – Karl Apr 18 '18 at 7:10
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Unfortunately, various definitions of Henry’s law and the corresponding proportionality constant $H$ or $K_\mathrm H$ exist. (For several definitions and corresponding parameter values, see Sander, R. Compilation of Henry’s law constants (version 4.0) for water as solvent. Atmos. Chem. Phys. 2015, 15, 4399–4981.) Therefore, it is important to identify the dimensions of the given data. In the question, the proportionality constant is given as Henry volatility $K_\mathrm H$ with

$$K_\mathrm H = 10^5\ \mathrm{atm}$$

Apparently,* since the given Henry volatility $K_\mathrm H$ is expressed in terms of pressure (the unit symbol “atm” stands for “standard atmosphere”, which is an obsolete unit of pressure; the use of this unit is actually deprecated), the used definition is

$$K_\mathrm H=\frac{p_{\ce{N2}}}{x_{\ce{N2}}}$$

where $p_{\ce{N2}}$ is partial pressure of nitrogen and $x_{\ce{N2}}$ is amount-of-substance fraction (the use of the unsystematic name “mole fraction” is deprecated) of nitrogen in the aqueous phase.

The amount-of-substance fraction $x_{\ce{N2}}$ is defined as

$$x_{\ce{N2}}=\frac{n_{\ce{N2}}}{n}$$

where $n_{\ce{N2}}$ is the amount of substance of nitrogen and $n$ is the total amount of substance. For dilute aqueous solutions, the total amount of substance is approximately equal to the amount of water

$$n\approx n_{\ce{H2O}}$$

which is given as $n_{\ce{H2O}}=10\ \mathrm{mol}$.

Thus, the amount of nitrogen $n_{\ce{N2}}$ can be calculated from the partial pressure of nitrogen $p_{\ce{N2}}$ as

$$\begin{aligned} n_{\ce{N2}}&=n\cdot x_{\ce{N2}}\\ &=n\cdot \frac{p_{\ce{N2}}}{K_\mathrm H}\\ &\approx n_{\ce{H2O}}\cdot \frac{p_{\ce{N2}}}{K_\mathrm H} \end{aligned}$$

For a mixture of gases, the partial pressure of nitrogen $p_{\ce{N2}}$ is defined as

$$p_{\ce{N2}}=x_{\ce{N2}}\cdot p$$

where $x_{\ce{N2}}$ is the amount-of-substance fraction of nitrogen in the gaseous phase (as apposed to the above-mentioned parameter in Henry’s law) and p is the total pressure.

Since the total pressure is given as $p=5\ \mathrm{atm}$ and the amount-of-substance fraction of nitrogen in air is given as $x_{\ce{N2}}=0.8$, the partial pressure of nitrogen in air is

$$\begin{aligned} p_{\ce{N2}}&=x_{\ce{N2}}\cdot p\\ &=0.8\times5\ \mathrm{atm}\\ &=4\ \mathrm{atm} \end{aligned}$$

* This use of the unit is in the question is actually not permissible. The unit symbol should not be used to provide specific information about the quantity and should never be the sole source of information on the quantity.

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  • $\begingroup$ I hate to say but I need to find the mole fraction of nitrogen in the solution. $\endgroup$ – Siddharth Panchal May 4 '16 at 12:24
  • $\begingroup$ @SiddharthPanchal I am not sure if I understand your problem with finding the amount-of-substance fraction of nitrogen in the solution. According to Henry’s law, $K_\mathrm H=\frac{p_{\ce{N2}}}{x_{\ce{N2}}}$; therefore, the amount-of-substance fraction of nitrogen in the solution is $x_{\ce{N2}}=\frac{p_{\ce{N2}}}{K_\mathrm H}=\frac{4\ \mathrm{atm}}{10^5\ \mathrm{atm}}=4\times10^{-5}$. In the end, however, the question is asking about the amount of substance $n_{\ce{N2}}$ and not about the amount-of-substance fraction $x_{\ce{N2}}$. $\endgroup$ – Loong May 4 '16 at 12:42
  • $\begingroup$ atm is a prefectly normal unit for pressure. Just because it's not SI is no reason to tell students to reject it. Also "mole fraction" is totally standard en.wikipedia.org/wiki/Mole_fraction , who says that term is deprecated and what's supposed to be misleading about it? $\endgroup$ – Karl Apr 18 '18 at 7:23

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