Question

Using Vein der Waal's equation, calculate the pressure exerted by 1 mole of carbon dioxide at \(0^{\circ} \mathrm{C}\) in a volume of (a) \(1.00\) liter, (b) \(0.05\) liter.

Short answer

The pressure exerted by 1 mole of CO2 at 0°C in a volume of 1.00 L is \(19.81\,atm\), and in a volume of \(0.05\,L\), it is approximately \(1725.21\,atm\).

Step-by-step solution

Convert the temperature to Kelvin

The given temperature is 0°C. We need to convert it to Kelvin for use in the equation. To do this, we add 273.15 to the Celsius temperature: \(T (K) = 0 + 273.15 = 273.15\,K\)

Calculate the pressure for a volume of 1.00 L

Now, we will calculate the pressure using the given volume of 1.00 L: 1. Substitute the values into the Van der Waals equation: \((P + a(\frac{n}{V})^2)(V - nb) = nRT\) 2. Plug in the values for CO2: \((P + 3.59(\frac{1}{1.00})^2)(1.00 - 0.0427(1)) = (1)(0.0821)(273.15)\) 3. Solve the equation: \((P + 3.59)(0.9573) = 22.41\) 4. Now, solve for P: \(P = \frac{22.41}{0.9573} - 3.59 = 19.81 \,atm\) Thus, the pressure exerted by 1 mole of CO2 at 0°C and 1.00 L is 19.81 atm.

Calculate the pressure for a volume of 0.05 L

Next, we will calculate the pressure using the given volume of 0.05 L: 1. Substitute the values into the Van der Waals equation: \((P + a(\frac{n}{V})^2)(V - nb) = nRT\) 2. Plug in the values for CO2: \((P + 3.59(\frac{1}{0.05})^2)(0.05 - 0.0427(1)) = (1)(0.0821)(273.15)\) 3. Solve the equation: \((P + 1436)(0.0073) = 22.41\) 4. Now, solve for P: \(P = \frac{22.41}{0.0073} - 1436 = 1725.21\,atm\) Thus, the pressure exerted by 1 mole of CO2 at 0°C and 0.05 L is approximately 1725.21 atm.

Key concepts

Van der Waals Equation

Understanding the Van der Waals equation is fundamental when dealing with real gases. Unlike ideal gases, real gases do not always follow the Ideal Gas Law perfectly due to their intermolecular forces and the volumes occupied by the gas particles themselves. The Van der Waals equation accounts for these factors by introducing two correction terms, 'a' and 'b'. Specifically, these terms correct for the intermolecular forces and finite volume of gas molecules, respectively.

The generalized form of the equation is \[ (P + \frac{a(n^2)}{V^2})(V - nb) = nRT \] where:\

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• \(P\) is the pressure of the gas,\
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• \(V\) is the volume,\
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• \(n\) is the number of moles,\
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• \(R\) is the universal gas constant,\
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• \(T\) is the temperature in Kelvin,\
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• \(a\) and \(b\) are specific to each gas, reflecting the strength of intermolecular forces and the volume excluded by the gas molecules.\
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This modified equation provides a more accurate prediction of gas behavior under non-ideal conditions, especially at high pressures and low volumes.

Pressure Calculation

Calculating the pressure of a gas involves understanding how particles within the gas interact and are confined. For ideal gases, the pressure can be calculated using the Ideal Gas Law. However, for real gases, pressure calculation is more complex and requires accounting for non-ideal behavior, as is the case with the Van der Waals equation. In this context, after accounting for the attractive forces between particles \(a(n/V)^2\) and the volume excluded by gas particles \(nb\), we can isolate the pressure \(P\) and solve the equation for it.

The steps for the pressure calculation using Van der Waals equation are as follows:\

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• Substitute the known values for \(n\), \(V\), \(R\), \(T\), \(a\), and \(b\) into the equation.
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• Perform the calculations to find the corrected pressure \(P\).
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Pressure calculation is crucial in predicting how gases will behave when subject to different temperatures and volumes, particularly in industrial applications where precision is key.

Molar Volume

Molar volume is a critical parameter often used in chemistry to describe the volume occupied by one mole of a substance (in this case, gas). The Standard Molar Volume of an ideal gas at Standard Temperature and Pressure (STP) is \(22.414 \, L/mol\). However, the molar volume of real gases deviates from this value due to particle interaction and particle size.

In the context of the Van der Waals equation, molar volume is explicitly considered. The term \(V - nb\), where \(b\) is a constant that represents the volume occupied by one mole of gas molecules, gives us an adjusted value that acknowledges the space taken up by the gas particles themselves. As such, molar volume is a variable directly relating to the equation and is essential when calculating the behavior of a real gas under different conditions.

Gas Laws

The Gas Laws are a set of laws that describe the behavior of gases in response to changes in temperature, volume, and pressure. The most well-known are Boyle's Law, Charles's Law, Avogadro's Law, and the Ideal Gas Law. These laws are combined in the Ideal Gas Law, which is expressed as \(PV = nRT\).

However, the Ideal Gas Law does not take into account the interactions between gas molecules or their volumes, which is where the Van der Waals equation comes into play as a more refined version tailored for real gases. This equation represents an advancement in our ability to model gas behavior more accurately in the real world, where gases exhibit non-ideal traits. Understanding these gas laws, both ideal and non-ideal, is a significant part of chemistry because they are foundational for predicting and explaining the properties and behavior of gases in various chemical reactions and processes.