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Chapter 14: Problem 55

At a given temperature and pressure, what mathematical relationship exists between the density and molar mass of a gas? Explain your answer.

Short Answer

Expert verified
Density is directly proportional to molar mass when temperature and pressure are constant, as per the equation \( \rho = \frac{PM}{RT} \).

Step by step solution

01

Define the Ideal Gas Law

The Ideal Gas Law is expressed as \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is temperature.
02

Rearrange for Moles (n)

The number of moles \( n \) can be expressed as \( \frac{m}{M} \), where \( m \) is the mass of the gas and \( M \) is the molar mass of the gas. Substitute this into the Ideal Gas Law to get \( PV = \frac{m}{M}RT \).
03

Solve for Density (ρ)

Density \( \rho \) is defined as mass per unit volume, or \( \rho = \frac{m}{V} \). Rearrange the equation from Step 2 to solve for \( m \) as \( m = \rho V \). Substitute into \( PV = \frac{m}{M}RT \), resulting in \( PV = \frac{\rho V}{M}RT \).
04

Simplify the Equation

Cancel out \( V \) from both sides of the equation to obtain \( P = \frac{\rho RT}{M} \).
05

Isolate Density (ρ)

Rearrange the equation from Step 4 to solve for \( \rho \), giving \( \rho = \frac{PM}{RT} \).
06

Express the Relationship Between Density and Molar Mass

The derived equation \( \rho = \frac{PM}{RT} \) shows the relationship between density \( \rho \) and molar mass \( M \). At constant pressure and temperature, density is directly proportional to molar mass.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Density
Density is one of those fundamental concepts in physics and chemistry that can sometimes seem perplexing; however, it can be easily understood with the right explanation. At its core, the density of a gas is a measure of how much mass it has within a certain volume. Mathematically, we express density as \( \rho = \frac{m}{V} \), where \( m \) is the mass and \( V \) is the volume.
Understanding density is particularly important when dealing with gases, which can change their density depending on a variety of factors, such as changes in pressure and temperature. If you increase the pressure of a gas while keeping the temperature constant, the gas particles get compressed, and the density increases since the same amount of mass takes up less space.
Similarly, if the temperature of a gas is increased at constant pressure, the particles spread out more due to increased kinetic energy, thereby decreasing its density as the same mass occupies a larger volume. In the Ideal Gas Law, we can see this manifestation where density \( \rho \) is linked with pressure \( P \), molar mass \( M \), constant \( R \), and temperature \( T \) via the equation \( \rho = \frac{PM}{RT} \). The higher the pressure or molar mass, the greater the density, and vice versa for temperature.
  • Density is mass per unit volume.
  • Affected by pressure and temperature changes.
  • Important for understanding gas behavior under different conditions.
Molar Mass
Molar mass is another crucial concept when working with gases and their properties. It's essentially the mass of one mole of a substance, usually expressed in grams per mole (g/mol). One mole contains Avogadro's number of molecules or atoms, which is approximately \( 6.022 \times 10^{23} \).
Why is molar mass important in the Ideal Gas Law? Well, it serves as a bridge connecting the number of moles to the actual substance's mass. In our Ideal Gas Law manipulation, we derive that \( n = \frac{m}{M} \), where \( n \) is the number of moles, \( m \) is the mass, and \( M \) is the molar mass. Substitute this into the Ideal Gas Law, and you see how molar mass directly influences other properties like density.
For gases, a higher molar mass means that, at a given pressure and temperature, the density will be higher because more massive molecules occupy a given volume. Therefore, understanding molar mass helps explain why different gases exhibit different densities under the same conditions.
  • Molar mass connects mass with moles.
  • Crucial for relating gas behavior to mass.
  • Directly affects gas density under specific conditions.
Temperature and Pressure
Temperature and pressure are vital factors affecting gas behavior and properties. Temperature is a measure of the average kinetic energy of gas molecules. As temperature increases, molecules move faster, and if the pressure doesn't change, they occupy a larger volume, making the gas less dense.
Pressure, on the other hand, is the force that the gas exerts against the walls of its container, usually measured in Pascals or atmospheres. An increase in pressure means that gas molecules are forced into a smaller space, increasing the density if the temperature remains constant.
The Ideal Gas Law, which is \( PV = nRT \), elegantly describes how these two variables are interwoven with volume and moles of gas. We see that with a constant amount of gas, increasing pressure will reduce volume and increase the density, while increasing temperature will increase volume and reduce density if pressure remains constant.
Understanding the impacts of temperature and pressure on gases helps predict their density and behavior in various scenarios, making this knowledge essential in scientific and engineering applications.
  • Temperature relates to kinetic energy of molecules.
  • Pressure is the force exerted by gas molecules.
  • Both affect gas density and behavior.

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Most popular questions from this chapter

Suppose that, for one semester, you can collect the following data on a random sample of college juniors and seniors for each class taken: a standardized final exam score, percentage of lectures attended, a dummy variable indicating whether the class is within the student's major, cumulative grade point average prior to the start of the semester, and SAT score. i. Why would you classify this data set as a cluster sample? Roughly, how many observations would you expect for the typical student? ii. Write a model, similar to equation (14.18) , that explains final exam performance in terms of attendance and the other characteristics. Use \(s\) to subscript student and \(c\) to subscript class. Which variables do not change within a student? iii. If you pool all of the data and use OLS, what are you assuming about unobserved student characteristics that affect performance and attendance rate? What roles do SAT score and prior GPA play in this regard? iv. If you think SAT score and prior GPA do not adequately capture student ability, how would you estimate the effect of attendance on final exam performance?

Consider the unobserved effects panel data model for a random draw \(i\), where \(a_{t}\) denotes different year intercepts: $$\begin{aligned}y_{i t}=& \alpha_{2} d_{2}+\ldots+\alpha_{T} d T+\beta_{1} x_{i t}+\beta_{2} x_{i t 2}^{2}+\gamma_{1} z_{i 1}+\gamma_{2} z_{i 2}+\gamma_{3} z_{i 2}^{2} \\\&+\gamma_{4} x_{i t} z_{i 1}+a_{i}+u_{i t}, t=1,2, \ldots, T\end{aligned}$$ Assuming a balanced panel, write down, for a given \(i\), the CRE equation that you would estimate using the RE estimator. Which parameter estimates should be identifical to the fixed effects estimates?

The density of nitrogen at 0.913 atm and \(18^{\circ} \mathrm{C}\) is \(1.07 \mathrm{g} / L .\) Explain how this shows that the formula of nitrogen is \(\mathrm{N}_{2}\) rather than just \(\mathrm{N}\).

Suppose that the idiosyncratic errors in \((14.4),\left\\{u_{i t}: t=1,2, \ldots, T\right\\},\) are serially uncorrelated with constant variance, \(\sigma_{u}^{2} .\) Show that the correlation between adjacent differences, \(\Delta u_{i t}\) and \(\Delta u_{i, t+1},\) is -.5. Therefore, under the ideal FE assumptions, first differencing induces negative serial correlation of a known value.

The data in CENSUS2000 is a random sample of individuals from the United States. Here we are interested in estimating a simple regression model relating the log of weekly income, Iweekinc , to schooling, educ. There are 29,501 observations. Associated with each individual is a state identifier (state) for the 50 states plus the District of Columbia. A less coarse geographic identifier is puma, which takes on 610 different values indicating geographic regions smaller than a state. Running the simple regression of lweekinc on \(e d u c\) gives a slope coefficient equal to. 1083 (to four decimal places). The heteroskedasticity- robust standard error is about. \(0024 .\) The standard error clustered at the puma level is about. \(.0027,\) and the standard error clustered at the state level is about.0033. For computing a confidence interval, which of these standard errors is the most reliable? Explain.
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