Question

It turns out that the van der Waals constant \(b\) equals four times the total volume actually occupied by the molecules of a mole of gas. Using this figure, calculate the fraction of the volume in a container actually occupied by Ar atoms (a) at STP, (b) at 200 atm pressure and \(0{ }^{\circ} \mathrm{C}\). (Assume for simplicity that the ideal-gas equation still holds.)

Short answer

The fraction of the volume in a container actually occupied by Ar atoms is approximately 0.000360 at STP and approximately 0.07207 at 200 atm pressure and 0°C.

Step-by-step solution

To calculate the volume of 1 mole of Ar at STP (Standard Temperature and Pressure), we need to know the values of P, n, R, and T. At STP, P = 1 atm and T = 273.15 K. The number of moles n is given as 1 mole, and the gas constant R = 0.0821 L atm / (K mol). Now we can substitute these values into the ideal gas equation: \(V = \frac{nRT}{P} = \frac{(1 \ \text{mol})(0.0821 \ \frac{\text{L atm}}{\text{K mol}})(273.15 \ \text{K})}{1 \ \text{atm}}\) #Step 2: Calculate the volume at STP#

Now calculate the volume: \(V = \frac{(1)(0.0821)(273.15)}{1}\) \(V = 22.414 \ \text{L}\) So, the volume of 1 mole of Ar at STP is 22.414 L. #Step 3: Calculate the fraction of the volume occupied by Ar atoms at STP#

We are given that the van der Waals constant b equals four times the total volume actually occupied by the molecules of a mole of gas. So, we can set up the equation: \(b = 4(\text{Volume occupied by Ar atoms})\) Rearrange the equation to solve for the volume occupied by Ar atoms: \(\text{Volume occupied by Ar atoms} = \frac{b}{4}\) For Ar, b = 0.0323 L/mol. So, we can substitute this value into the equation: \(\text{Volume occupied by Ar atoms} = \frac{0.0323 \ \text{L/mol}}{4}\) \(\text{Volume occupied by Ar atoms} = 0.008075 \ \text{L/mol}\) Now, we can find the fraction of the volume in a container actually occupied by Ar atoms at STP using the following equation: \(\text{Fraction of volume occupied} = \frac{\text{Volume occupied by Ar atoms}}{\text{Volume at STP}}\) \(\text{Fraction of volume occupied} = \frac{0.008075}{22.414}\) #Step 4: Calculate the fraction of the volume occupied by Ar atoms at STP#

Now calculate the fraction: \(\text{Fraction of volume occupied} = \frac{0.008075}{22.414}\) \(\text{Fraction of volume occupied} = 0.000360\) So, the fraction of the volume in a container actually occupied by Ar atoms at STP is approximately 0.000360. #Step 5: Calculate the volume of 1 mole of Ar at 200 atm pressure and 0°C#

First, we will convert 0°C to Kelvin: T = 0 + 273.15 = 273.15 K Now, we will calculate the volume of 1 mole of Ar at 200 atm pressure and 273.15 K using the ideal gas equation, with P = 200 atm and n, R, and T as previously determined: \(V = \frac{nRT}{P} = \frac{(1)(0.0821)(273.15)}{200}\) #Step 6: Calculate the volume at 200 atm and 0°C#

Now calculate the volume: \(V = \frac{(1)(0.0821)(273.15)}{200}\) \(V = 0.11207 \ \text{L}\) So, the volume of 1 mole of Ar at 200 atm pressure and 0°C is 0.11207 L. #Step 7: Calculate the fraction of the volume occupied by Ar atoms at 200 atm pressure and 0°C#

Using the previously determined value for the volume occupied by Ar atoms (0.008075 L/mol), we can find the fraction of the volume in a container actually occupied by Ar atoms at 200 atm pressure and 0°C, using the following equation: \(\text{Fraction of volume occupied} = \frac{\text{Volume occupied by Ar atoms}}{\text{Volume at 200 atm and 0°C}}\) \(\text{Fraction of volume occupied} = \frac{0.008075}{0.11207}\) #Step 8: Calculate the fraction of the volume occupied by Ar atoms at 200 atm pressure and 0°C#

Now calculate the fraction: \(\text{Fraction of volume occupied} = \frac{0.008075}{0.11207}\) \(\text{Fraction of volume occupied} = 0.07207\) So, the fraction of the volume in a container actually occupied by Ar atoms at 200 atm pressure and 0°C is approximately 0.07207. In conclusion, the fraction of the volume in a container actually occupied by Ar atoms: (a) at STP is approximately 0.000360 (b) at 200 atm pressure and 0°C is approximately 0.07207.

Key concepts

Ideal Gas Equation

Understanding the ideal gas equation is fundamental when dealing with problems concerning gases under certain conditions. This equation is usually written as
\( PV = nRT \),
where P represents the pressure of the gas, V is the volume, n denotes the number of moles of gas, R is the ideal gas constant, and T is the temperature in Kelvin. The ideal gas equation is a cornerstone in physics and chemistry that describes the behavior of an ideal or perfect gas.

This equation assumes that the gas particles are point masses with no significant volume and that there are no interactive forces between the particles, except during elastic collisions. This model is often a close approximation of real gas behavior under conditions of high temperature and low pressure. It is also useful when calculating how the volume and pressure or other properties of a gas will change in response to a change in temperature or the number of gas moles. However, for real gases under high pressure or low temperature, deviations from the ideal gas law are often observed.

Molar Volume at STP

Molar volume is a property of gases that refers to the volume one mole of a substance occupies at specified conditions of temperature and pressure. At standard temperature and pressure (STP), which is 0 degrees Celsius and 1 atmosphere pressure, the molar volume of any ideal gas is 22.414 liters per mole. This value comes from the ideal gas law when substituted with the conditions at STP,
\( V = \frac{nRT}{P} \).

In practice, the molar volume at STP provides a handy reference for calculating the volume of gases because it is a constant for all ideal gases. This standardization allows for comparisons between different gases and simplifies calculations in many chemical and physical equations. Nevertheless, it's important to keep in mind that this value is an approximated one, derived from the behavior of an ideal gas, which means real gases may not exactly occupy 22.414 L/mol at STP due to interactions between molecules and the volume those molecules occupy.

Fraction of Volume Occupied by Gas

When examining gases, it's not only important to understand their behavior under different conditions but also to consider how much volume the gas particles actually take up. This includes the space filled by the gas particles themselves, as opposed to the space within which the particles are free to move – an aspect often neglected in the ideal gas model.

The fraction of volume occupied by a gas within a certain volume of space can be quantified. According to van der Waals, the volume occupied by the molecules of a mole of gas is reflected in the constant b, specifically as \( b = 4(\text{Volume occupied by gas molecules}) \). This provides a correction to the ideal gas law for the volume actually occupied by the gas. We can calculate the fraction of volume occupied by determining the ratio of the volume occupied by the gas molecules to the total volume. This calculation illustrates that even under standard conditions, the fraction occupied is quite small, with the molecules taking up only a minor portion of the total volume, while under high pressure the fraction significantly increases. This consideration is vital for understanding real gas behavior, especially under conditions far removed from the idealized models.