Chapter 10, Problem 107SCQ

A 1.0-L flask contains 10.0 g each of O₂ and CO₂ at 25 ^°C.

1. (a) Which gas has the greater partial pressure, O₂ or CO₂, or are they the same?
2. (b) Which molecules have the greater rms speed, or are they the same?
3. (c) Which molecules have the greater average kinetic energy, or are they the same?

Interpretation Introduction

Interpretation:

For the given set of gases under given temperature, volume and amount the gas with greater partial pressure, with greater $\text{rms}$ speed and the gas with greater average kinetic energy should be determined.

Concept introduction:

Ideal gas Equation:

Any gas is described by using four terms namely pressure, volume, temperature and the amount of gas.  Thus combining three laws namely Boyle’s, Charles’s Law and Avogadro’s Hypothesis the following equation could be obtained.  It is referred as ideal gas equation.

  $\begin{array}{l}\text{V }\propto \frac{\text{nT}}{\text{P}}\\ \text{V = R}\frac{\text{nT}}{\text{P}}\\ \text{PV = nRT}\\ \text{where,}\\ \text{n = moles of gas}\\ \text{P = pressure}\\ \text{T = temperature}\\ \text{R = gas }\text{constant}\end{array}$

Under some conditions gases don not behave like ideal gas that is they deviate from their ideal gas properties.  At lower temperature and at high pressures the gas tends to deviate and behave like real gases.

Boyle’s Law:

At given constant temperature conditions the mass of given ideal gas in inversely proportional to its volume.

Charles’s Law:

At given constant pressure conditions the volume of ideal gas is directly proportional to the absolute temperature.

Avogadro’s Hypothesis:

Two equal volumes of gases with same temperature and pressure conditions tend to have same number of molecules with it.

The root mean square velocity $\mu$ is defined as the measure of velocity of particle in gas.  It is the method to determine the single velocity value for particles.

Root mean square velocity can be determined,

  ${\mu }_{rms}={\left(\frac{3RT}{M}\right)}^{1/2}$ (1)

  $\begin{array}{l}\left(\text{gas constant}\right)R=\frac{8.314J}{Kmol}\\ M=Molarmass\end{array}$

Molar mass: The molar mass of a substance is determined by dividing the given mass of substance by the amount of the substance.

Average Kinetic energy: The kinetic energy for the gas is directly proportional to the kelvin temperature.  The kinetic energy is equal to half of the multiplied value obtained by multiplication of mass of gas with $\text{rms}$ velocity of the gas.

Answer to Problem 107SCQ

The gas ${\text{O}}_{\text{2}}$ has greater partial pressure than the other gas ${\text{CO}}_{\text{2}}$

Explanation of Solution

Given:

  $\begin{array}{l}\text{Volume,}\text{V=1L}\\ \text{Mass}\text{of}{\text{O}}_{\text{2}}\text{=10g}\\ \text{moles =}\frac{\text{mass}}{\text{molar mass}}=\frac{10g}{\text{32g/mol}}=0.3125mol\\ \text{Mass}\text{of}{\text{CO}}_{\text{2}}\text{=10g}\\ \text{moles =}\frac{\text{mass}}{\text{molar mass}}=\frac{10g}{\text{44}\text{.01g/mol}}=0.2272mol\\ \text{Temperature,}{\text{T = 25}}^{\text{o}}\text{C = 273}\text{.15+25 = 298}\text{.15K}\end{array}$

Using ideal gas equation the partial pressure for each of the given gas is calculated as follows,

  $\begin{array}{l}\text{PV= nRT}\\ \text{P = }\frac{\text{nRT}}{\text{V}}\\ {\text{P}}_{{\text{O}}_{\text{2}}}\text{=}\frac{\text{0}\text{.3125mol×0}\text{.0821×298}\text{.15K}}{\text{1}}\text{=}\text{7}\text{.65atm}\\ {\text{P}}_{{\text{CO}}_{\text{2}}}\text{=}\frac{\text{0}\text{.2272mol×0}\text{.0821×298}\text{.15K}}{\text{1}}\text{=}\text{5}\text{.56atm}\end{array}$

From the above calculation it is clear that $O_2$ has larger partial pressure it is due to the fact that $O_2$ have larger number of molecules compared with carbon dioxide which is obtained by multiplying moles with $\text{6}{\text{.023×10}}^{\text{23}}$.

Interpretation Introduction

Interpretation: For the given set of gases under given temperature, volume and amount the gas with greater partial pressure, with greater $\text{rms}$ speed and the gas with greater average kinetic energy should be determined.

Concept introduction:

Ideal gas Equation:

Any gas is described by using four terms namely pressure, volume, temperature and the amount of gas.  Thus combining three laws namely Boyle’s, Charles’s Law and Avogadro’s Hypothesis the following equation could be obtained.  It is referred as ideal gas equation.

  $\begin{array}{l}\text{V }\propto \frac{\text{nT}}{\text{P}}\\ \text{V = R}\frac{\text{nT}}{\text{P}}\\ \text{PV = nRT}\\ \text{where,}\\ \text{n = moles of gas}\\ \text{P = pressure}\\ \text{T = temperature}\\ \text{R = gas }\text{constant}\end{array}$

Under some conditions gases don not behave like ideal gas that is they deviate from their ideal gas properties.   At lower temperature and at high pressures the gas tends to deviate and behave like real gases.

Boyle’s Law:

At given constant temperature conditions the mass of given ideal gas in inversely proportional to its volume.

Charles’s Law:

At given constant pressure conditions the volume of ideal gas is directly proportional to the absolute temperature.

Avogadro’s Hypothesis:

Two equal volumes of gases with same temperature and pressure conditions tend to have same number of molecules with it.

The root mean square velocity $\mu$ is defined as the measure of velocity of particle in gas.  It is the method to determine the single velocity value for particles.

Root mean square velocity can be determined,

  ${\mu }_{rms}={\left(\frac{3RT}{M}\right)}^{1/2}$ (1)

  $\begin{array}{l}\left(\text{gas constant}\right)R=\frac{8.314J}{Kmol}\\ M=Molarmass\end{array}$

Molar mass: The molar mass of a substance is determined by dividing the given mass of substance by the amount of the substance.

Average Kinetic energy: The kinetic energy for the gas is directly proportional to the kelvin temperature.  The kinetic energy is equal to half of the multiplied value obtained by multiplication of mass of gas with $\text{rms}$ velocity of the gas.

Answer to Problem 107SCQ

The gas ${\text{O}}_{\text{2}}$ has greater $\text{rms}$ speed.

Explanation of Solution

The $\text{rms}$ speed for the given set of gases is determined as follows,

  $\begin{array}{l}{\text{μ}}_{\text{rms}}\text{=}{\left(\frac{\text{3}\text{R}\text{T}}{\text{M}}\right)}^{\text{1/2}}\\ {\text{rms speed for O}}_{\text{2}}\text{= }{\left(\frac{\text{38}\text{.314}\times \text{298}\text{.15}}{32}\right)}^{\text{1/2}}={\left(\frac{11423.32}{32}\right)}^{\text{1/2}}=356.98m/s\\ {\text{rms speed for CO}}_{\text{2}}\text{= }{\left(\frac{\text{38}\text{.314}\times \text{298}\text{.15}}{44.01}\right)}^{\text{1/2}}={\left(\frac{11423.32}{44.01}\right)}^{\text{1/2}}=259.6m/s\end{array}$

From the above calculation it is clear that $O_2$ has larger $\text{rms}$ speed value.

Interpretation Introduction

Interpretation: For the given set of gases under given temperature, volume and amount the gas with greater partial pressure, with greater $\text{rms}$ speed and the gas with greater average kinetic energy should be determined.

Concept introduction:

Ideal gas Equation:

Any gas is described by using four terms namely pressure, volume, temperature and the amount of gas.  Thus combining three laws namely Boyle’s, Charles’s Law and Avogadro’s Hypothesis the following equation could be obtained.  It is referred as ideal gas equation.

  $\begin{array}{l}\text{V }\propto \frac{\text{nT}}{\text{P}}\\ \text{V = R}\frac{\text{nT}}{\text{P}}\\ \text{PV = nRT}\\ \text{where,}\\ \text{n = moles of gas}\\ \text{P = pressure}\\ \text{T = temperature}\\ \text{R = gas }\text{constant}\end{array}$

Under some conditions gases don not behave like ideal gas that is they deviate from their ideal gas properties.  At lower temperature and at high pressures the gas tends to deviate and behave like real gases.

Boyle’s Law:

At given constant temperature conditions the mass of given ideal gas in inversely proportional to its volume.

Charles’s Law:

At given constant pressure conditions the volume of ideal gas is directly proportional to the absolute temperature.

Avogadro’s Hypothesis:

Two equal volumes of gases with same temperature and pressure conditions tend to have same number of molecules with it.

The root mean square velocity $\mu$ is defined as the measure of velocity of particle in gas. It is the method to determine the single velocity value for particles.

Root mean square velocity can be determined,

  ${\mu }_{rms}={\left(\frac{3RT}{M}\right)}^{1/2}$ (1)

  $\begin{array}{l}\left(\text{gas constant}\right)R=\frac{8.314J}{Kmol}\\ M=Molarmass\end{array}$

Molar mass: The molar mass of a substance is determined by dividing the given mass of substance by the amount of the substance.

Average Kinetic energy: The kinetic energy for the gas is directly proportional to the kelvin temperature.  The kinetic energy is equal to half of the multiplied value obtained by multiplication of mass of gas with $\text{rms}$ velocity of the gas.

Answer to Problem 107SCQ

Both the given gases have same kinetic energy since both are under same temperature conditions.

Explanation of Solution

The kinetic energy for the molecules is determined by the temperature in which the gases are placed.  Both the given gases are placed under same temperature that is at ${\text{25}}^{\text{o}}\text{C}$ tends to have equal kinetic energy values.