If equal mols of $N_{2}$ $"N"_2$ and $Ar$ $"Ar"$ are in the same container with a total pressure of $10 atm$ $"10 atm"$ , $(A)$ $(A)$ what are their partial pressures? $(B)$ $(B)$ what is the effusion speed of $H_{2}$ $"H"_2$ if that of $He$ $"He"$ is $600 m/s$ $"600 m/s"$ at a certain temperature?

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If equal mols of $N_{2}$ $"N"_2$ and $Ar$ $"Ar"$ are in the same container with a total pressure of $10 atm$ $"10 atm"$ , $(A)$ $(A)$ what are their partial pressures? $(B)$ $(B)$ what is the effusion speed of $H_{2}$ $"H"_2$ if that of $He$ $"He"$ is $600 m/s$ $"600 m/s"$ at a certain temperature?

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Answer 1 · 2017-06-04T23:38:31

$A)$ $A)$ $5 atm$ $"5 atm"$ each.

$B)$ $B)$ The lighter gas $B$ $B$ effuses at a speed $v_{B}$ $v_B$ that is $\sqrt{\frac{M_{A}}{M_{B}}}$ $sqrt(M_A/M_B)$ times as fast as $v_{A}$ $v_A$ , where $M_{A}$ $M_A$ is the molar mass of $A$ $A$ .

$A)$ $A)$ If the container is filled with an equal number of each particle, then we know that:

$n_{N_{2}} = n_{A r}$ $n_(N_2) = n_(Ar)$

When this is the case, assuming ideal gases, their partial pressures should be identical.

Mathematically, this results in the following mol fractions:

$χ_{N_{2}} = \frac{n_{N_{2}}}{n_{A r} + n_{N_{2}}}$ $chi_(N_2) = (n_(N_2))/(n_(Ar) + n_(N_2))$

$χ_{A r} = \frac{n_{A r}}{n_{A r} + n_{N_{2}}}$ $chi_(Ar) = (n_(Ar))/(n_(Ar) + n_(N_2))$

But since $n_{N_{2}} = n_{A r}$ $n_(N_2) = n_(Ar)$ , we expect $χ_{N_{2}} = χ_{A r} = 0.5$ $chi_(N_2) = chi_(Ar) = 0.5$ . Let's just say we had $1 mol$ $"1 mol"$ of each gas. Then:

$χ_{N_{2}} = \frac{1}{1 + 1} = 0.5 = χ_{A r}$ $chi_(N_2) = 1/(1 + 1) = 0.5 = chi_(Ar)$

Assuming ideal gases, we can show that both gases have the same partial pressure, the pressure exerted by each gas in the mixture:

$P_{N_{2}} = χ_{N_{2}} P_{t o t}$ $color(blue)(P_(N_2)) = chi_(N_2)P_(t ot)$

$= χ_{A r} P_{t o t} = P_{A r}$ $= chi_(Ar)P_(t ot) = color(blue)(P_(Ar))$

$= 0.5 (10 atm) = 5 atm$ $= 0.5("10 atm") = color(blue)("5 atm")$

$B)$ $B)$ I'm not sure what part $B$ $B$ has to do with part $A$ $A$ . I will assume the question means to write $Ar$ $"Ar"$ and $N_{2}$ $"N"_2$ instead of $He$ $"He"$ and $H_{2}$ $"H"_2$ .

We are given, then, that argon effuses at $600 m/s$ $"600 m/s"$ . Effusion can be derived from the expression for some sort of speed for each gas. The root-mean-square speed equation is fine:

$v_{r m s} = \sqrt{\frac{3 R T}{M}}$ $v_(rms) = sqrt((3RT)/M)$

where $R$ $R$ and $T$ $T$ are known from the ideal gas law, and $M$ $M$ is the molar mass in $kg/mol$ $"kg/mol"$ .

The rate of effusion, $z_{e f f}$ $z_(eff)$ is proportional to the speed $v$ $v$ , so the proportionality constants cancel out in a ratio:

$\frac{z_{e f f, A r}}{z_{e f f, N_{2}}} = \frac{v_{A r}}{v_{N_{2}}} = \sqrt{\frac{M_{N_{2}}}{M_{A r}}}$ $z_(eff,Ar)/(z_(eff,N_2)) = bb(v_(Ar)/(v_(N_2))) = bb(sqrt(M_(N_2)/(M_(Ar))))$

The relationship of $\frac{z_{e f f, i}}{z_{e f f, j}}$ $(z_(eff,i))/(z_(eff,j))$ to $\sqrt{\frac{M_{j}}{M_{i}}}$ $sqrt(M_j/M_i)$ is known as Graham's law of effusion, but we will instead be using the relationship with $\frac{v_{i}}{v_{j}}$ $v_i/v_j$ , the ratio of the speeds.

We are given that $v_{A r} = 600 m/s$ $v_(Ar) = "600 m/s"$ , so:

$v_{N_{2}} = v_{A r} \sqrt{\frac{M_{A r}}{M_{N_{2}}}}$ $color(blue)(v_(N_2)) = v_(Ar) sqrt(M_(Ar)/M_(N_2))$

$= 600 m/s \cdot \sqrt{\frac{0.039948 kg/mol}{0.028014 kg/mol}}$ $= "600 m/s" cdot sqrt("0.039948 kg/mol"/"0.028014 kg/mol")$

$=$ $=$ $716.5 m/s$ $color(blue)("716.5 m/s")$

This should make sense, that the lighter gas, $N_{2}$ $"N"_2$ , effused faster.

If it really is supposed to be $He$ $"He"$ and $H_{2}$ $"H"_2$ , then you should expect:

$v_{H_{2}} = v_{H e} \sqrt{\frac{M_{H e}}{M_{H_{2}}}}$ $color(blue)(v_(H_2)) = v_(He) sqrt(M_(He)/M_(H_2))$

$= 600 m/s \cdot \sqrt{\frac{0.0040026 kg/mol}{0.0020158 kg/mol}}$ $= "600 m/s" cdot sqrt("0.0040026 kg/mol"/"0.0020158 kg/mol")$

$=$ $=$ $845.5 m/s$ $color(blue)("845.5 m/s")$

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If equal mols of $N_{2}$ $"N"_2$ and $Ar$ $"Ar"$ are in the same container with a total pressure of $10 atm$ $"10 atm"$ , $(A)$ $(A)$ what are their partial pressures? $(B)$ $(B)$ what is the effusion speed of $H_{2}$ $"H"_2$ if that of $He$ $"He"$ is $600 m/s$ $"600 m/s"$ at a certain temperature?

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If equal mols of N2"N"_2 and Ar"Ar" are in the same container with a total pressure of 10 atm"10 atm", (A)(A) what are their partial pressures? (B)(B) what is the effusion speed of H2"H"_2 if that of He"He" is 600 m/s"600 m/s" at a certain temperature?

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If equal mols of $N_{2}$ $"N"_2$ and $Ar$ $"Ar"$ are in the same container with a total pressure of $10 atm$ $"10 atm"$ , $(A)$ $(A)$ what are their partial pressures? $(B)$ $(B)$ what is the effusion speed of $H_{2}$ $"H"_2$ if that of $He$ $"He"$ is $600 m/s$ $"600 m/s"$ at a certain temperature?