Problem 69 - University Physics with Modern Physics

Question

In the ideal-gas equation, the number of moles per volume \(n / V\) is simply equal to \(p / R T\) . In the van der Waals equation, solving for \(n / V\) in terms of the pressure \(p\) and temperature \(T\) is somewhat more involved. (a) Show the van der Waals equation can be written as $$\frac{n}{V}=\left(\frac{p+a n^{2} / V^{2}}{R T}\right)\left(1-\frac{b n}{V}\right)$$ (b) The van der Waals parameters for hydrogen sulfide gas \(\left(\mathrm{H}_{2} \mathrm{S}\right)\) are \(a=0.448 \mathrm{J} \cdot \mathrm{m}^{3} / \mathrm{mol}^{2}\) and \(b=4.29 \times 10^{-5} \mathrm{m}^{3} / \mathrm{mol}\) . Determine the number of moles per volume of \(\mathrm{H}_{2} \mathrm{S}\) gas at \(127^{\circ} \mathrm{C}\) and an absolute pressure of \(9.80 \times 10^{5} \mathrm{Pa}\) as follows: (i) Calculate a first approximation using the ideal-gas equation, \(n / V=p / R T\) . (ii) Substitute this approximation for \(n / V\) into the right-hand side of the equation in part (a). The result is a new, improved approximation for \(n / V\) . (iii) Substitute the new approximation for \(n / V\) into the right-hand side of the equation in (a). The result is a further improved approximation for \(n / V\) . (iv) Repeat step (iii) until successive approximations agree to the desired level of accuracy (in this case, to three significant figures). (c) Compare your final result in part (b) to the result \(p / R T\) obtained using the ideal-gas equation. Which result gives a larger value of \(n / V ?\)

Step-by-Step Solution

Answer

The van der Waals equation gives a larger value of \( n/V \) than the ideal gas equation.

Step 1: Rewrite van der Waals Equation

Start with the van der Waals equation: \[ p + a\left(\frac{n}{V}\right)^2 = \frac{nRT}{V - nb} \]. Multiply both sides by the denominator \((V - nb)\) to eliminate the fraction.

Step 2: Expand and Rearrange

Expanding gives: \[ p(V - nb) = nRT - a\left(\frac{n}{V}\right)^2(V - nb) \]. Rearrange to solve for \( \frac{n}{V} \). This can be expressed as: \[ \frac{n}{V} = \left(\frac{p + a\left(\frac{n}{V}\right)^2}{RT}\right)\left(1 - \frac{bn}{V}\right) \].

Step 3: Convert Temperature to Kelvin

Convert the given temperature from Celsius to Kelvin. Temperature, \( T = 127^{\circ} \text{C} + 273.15 = 400.15 \text{K} \).

Step 4: First Approximation Using the Ideal Gas Equation

Using the ideal gas approximation \( \frac{n}{V} = \frac{p}{RT} \) with \( R = 8.314 \text{ J/(mol·K)} \) and pressure \( p = 9.80 \times 10^5 \text{ Pa} \), first calculate: \[ \frac{n}{V} = \frac{9.80 \times 10^5}{8.314 \times 400.15}\]. This gives the first approximation for \( \frac{n}{V} \).

Step 5: Substitute First Approximation into van der Waals Equation

Substitute the first approximation into the right-hand side of the equation from part (a): \[ \frac{n}{V} \approx \left(\frac{9.80 \times 10^5 + 0.448\left(\frac{n}{V}\right)^{2}}{8.314 \times 400.15}\right)\left(1 - 4.29 \times 10^{-5}\left(\frac{n}{V}\right)\right) \]. Calculate the new approximation.

Step 6: Repeat Substitution for Improved Approximation

Substitute the new approximation back into the formula. Iterate this process until two successive values of \( \frac{n}{V} \) differ by less than the desired precision. Continue calculating until the value stabilizes to three significant figures.

Step 7: Compare Values of n/V

Finally, compare the result obtained using the van der Waals equation to \( \frac{n}{V} = \frac{p}{RT} \). Determine which one gives a larger \( \frac{n}{V} \).

Key Concepts

Ideal Gas Law

The ideal gas law is a fundamental equation in physics and chemistry that describes the behavior of an ideal gas. It is expressed as \( PV = nRT \), where:

• \( P \) stands for pressure
• \( V \) is the volume
• \( n \) is the number of moles
• \( R \) represents the ideal gas constant
• \( T \) is the absolute temperature in Kelvin

This equation provides an approximation of the state of a gas under certain conditions. It is based on the assumption that the gas particles are point masses with no volume and no interactions between them. However, in reality, gases have intermolecular forces and occupy space, which can lead to deviations from ideal behavior under high pressure or low temperature conditions.
The ideal gas law is often used because of its simplicity and is a good approximation in many practical situations.

Gas Constant

The gas constant, denoted by \( R \), is a crucial factor in equations like the ideal gas law. It has a consistent value that applies universally, making it invaluable for calculations involving gases. The value of the ideal gas constant is approximately \( 8.314 \text{ J/(mol·K)} \), though it can be found in different units depending on the equation's requirements, such as \( ext{L·atm/(mol·K)} \).
The constancy of \( R \) allows scientists to predict the behavior of gases with a high degree of accuracy under a range of conditions. Since it is used in conjunction with temperature, pressure, and volume, \( R \) acts to bridge these variables, ensuring they remain consistent across different scenarios.

Temperature Conversion

Temperature conversion is a critical step in gas law calculations since most equations require temperatures to be in Kelvin, rather than Celsius or Fahrenheit. Kelvin is the absolute temperature scale, which ensures all temperatures used in equations are positive, preventing mathematical errors.
To convert Celsius to Kelvin, you simply add 273.15 to the Celsius temperature. For example:

• If the temperature is \( 127^{\circ} \text{C} \), convert it as follows: \( 127 + 273.15 = 400.15 \text{ K} \).

This simple addition accounts for the starting point of the Kelvin scale, based on absolute zero, making it a straightforward yet impactful process in gas solubility and other thermodynamic calculations.

Approximations in Physics

Approximations play a pivotal role in physics, allowing scientists to make simplified calculations that are useful despite minor inaccuracies. In the context of the ideal gas law and the van der Waals equation, approximations help manage the complexity of real gas behaviors.
For instance, using the ideal gas law \(\frac{n}{V} = \frac{p}{RT} \) as a first approximation allows for quick estimations of gas properties. However, more sophisticated models like the van der Waals equation refine these calculations by incorporating parameters \( a \) and \( b \), which adjust for intermolecular forces and the finite volume of gas particles. This iterative process of approximation converges toward a more accurate depiction of the system.
Such methods, though initially approximate, ultimately provide a closer match to observed phenomena, illustrating the balance between theoretical prediction and practical observation in physics.