The density of a gas of unknown molar mass was measured as a function of pressure at 0 C, as in the table that follows. (a) Determine a precise molar mass for the gas. [Hint: Graph d>P versus P.]

Verified step by step guidance

1. Step 1: The first step is to understand the relationship between the density (d) of a gas, its molar mass (M), and the pressure (P). According to the ideal gas law, PV = nRT, where n is the number of moles, R is the gas constant, and T is the temperature. We can rearrange this equation to express n (the number of moles) as P/RT. Since density is mass/volume, we can express it as (n*M)/V, where M is the molar mass. Substituting n from the ideal gas law, we get d = PM/RT.
2. Step 2: From the equation in step 1, we can see that the density of a gas is directly proportional to its pressure and molar mass, and inversely proportional to the temperature. This means that if we plot the density of the gas against its pressure, the slope of the line will be equal to the molar mass divided by the gas constant times the temperature (M/RT).
3. Step 3: Plot the given data of density versus pressure on a graph. The slope of the line that best fits the data points will give you the value of M/RT.
4. Step 4: To find the molar mass, you need to multiply the slope by RT. Since the temperature is given as 0 C, you need to convert it to Kelvin by adding 273.15. The value of R (the gas constant) is 0.0821 L.atm/(mol.K) in these units.
5. Step 5: Multiply the slope by RT to get the molar mass of the gas. This will give you the precise molar mass of the gas in g/mol.

Key Concepts

Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of a gas through the equation PV = nRT. This law is fundamental in understanding gas behavior and allows for the calculation of molar mass when density and pressure are known. It provides a framework for analyzing how changes in one variable affect others in a gas system.

Density of Gases

Density is defined as mass per unit volume and for gases can be expressed as d = PM/RT, where P is pressure, M is molar mass, R is the ideal gas constant, and T is temperature. Understanding this relationship is crucial for determining the molar mass of a gas from its density measurements. The density of a gas can change with pressure and temperature, making it essential to consider these factors in calculations.

Graphing Relationships

Graphing the relationship between density and pressure, specifically plotting d/P versus P, allows for the visualization of how these two variables interact. This method can reveal linear relationships that can be used to derive the molar mass of the gas. Understanding how to interpret graphs is essential for analyzing experimental data and drawing conclusions from it.