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Chapter 14: Problem 19

ssm What is the density (in \(\mathrm{kg} / \mathrm{m}^{3} )\) of nitrogen gas (molecular mass \(=28 \mathrm{u}\) ) at a pressure of 2.0 atmospheres and a temperature of 310 \(\mathrm{K} ?\)

Short Answer

Expert verified
The density of nitrogen gas is approximately 61.80 kg/m鲁.

Step by step solution

01

Recall Ideal Gas Law

The ideal gas law is given by the equation \[PV = nRT\]where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(T\) is the temperature.
02

Rearrange for Moles

Rearrange the ideal gas law to find the number of moles:\[n = \frac{PV}{RT}\]
03

Find Moles per Volume

We want the moles per volume, equivalent to the density as moles per volume.\[ \frac{n}{V} = \frac{P}{RT} \]
04

Calculate Density in kg/m鲁

Density \( \rho \) in kg/m鲁 is given by the equation:\[ \rho = \frac{P \cdot M}{RT} \]where \(M\) is the molar mass in kg/mol. Convert molecular mass from atomic mass units to kilograms:\[M = 28 \times 10^{-3} \ \mathrm{kg/mol} \]
05

Insert Values

Insert the given values into the density equation:- Pressure \(P = 2.0\ \text{atm} = 2.0 \times 101325\ \text{Pa}\)- Molar mass \(M = 28 \times 10^{-3} \ \mathrm{kg/mol} \)- Gas constant \(R = 8.314\ \text{J/mol鈰匥}\)- Temperature \(T = 310\ \text{K}\)\[ \rho = \frac{(2.0 \times 101325) \times (28 \times 10^{-3})}{8.314 \times 310} \]
06

Compute Numeric Value

Calculate the density:\[ \rho = \frac{5686200 \times 0.028}{2577.34} = \frac{159213.6}{2577.34} \approx 61.80\ \mathrm{kg/m}^3 \]

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Key Concepts

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

Density Calculation
Density is a measure of how much mass is present in a unit volume of a substance. For gases, it can be calculated using the ideal gas law. To find the density \( \rho \) of a gas, we need to consider its mass and volume at a given temperature and pressure.

The equation we use is derived from the ideal gas law itself and is given by:
  • \( \rho = \frac{P \cdot M}{RT} \)

Here,
  • \( P \) is the pressure in Pascals (\( \text{Pa} \))
  • \( M \) is the molar mass in kilograms per mole (\( \text{kg/mol} \))
  • \( R \) is the gas constant
  • \( T \) is the temperature in Kelvin (\( \text{K} \))

The density will be calculated in units of \( \text{kg/m}^3 \), which is a standard SI unit used for density.
Molecular Mass
Molecular mass, or molar mass, is crucial for calculating gas properties like density. It represents the weight of one mole of a substance, usually expressed in atomic mass units (\( \ ext{u} \)).

To use molecular mass in the gas law calculations, you should convert it to kilograms per mole (\( \ ext{kg/mol} \)). For nitrogen gas with a molecular mass of 28 \( \ ext{u} \), the conversion is straightforward:
  • Multiply by \( 10^{-3} \) to get \( M = 28 \times 10^{-3} \ \text{kg/mol} \)

This conversion is necessary as the ideal gas law employs SI units. Make sure the molecular mass is converted from units of grams per mole to kilograms per mole for consistency with other SI units in physical calculations.
Pressure and Temperature
Pressure and temperature greatly affect the behavior of gases. In the context of the ideal gas law, both are fundamental parameters.

Pressure:\Typically measured in atmospheres (\( \ ext{atm} \)) or Pascals (\( \ ext{Pa} \)), it's important to convert everything to Pascals for the ideal gas law:
  • 1 \( \ ext{atm} = 101325 \ ext{Pa} \)

Temperature:\It should always be in Kelvin (\( \ ext{K} \)) when using the ideal gas law. Convert Celsius to Kelvin by adding 273.15.

These conversions ensure internal consistency within the equations and allow accurate calculation of properties like density using the ideal gas law.
Gas Constant
The ideal gas constant \( R \) is a key part of the ideal gas law equation. It ties together pressure, volume, temperature, and amount of substance (moles).

\( R \) has a fixed value:
  • \( 8.314 \ \ ext{J/mol鈰匥} \)

This unit \( \ ext{J/mol鈰匥} \) shows that \( R \) relates energy for each mole of substance at every Kelvin. The value allows us to balance the equation:
  • \( PV = nRT \)

By maintaining consistent units across all factors (temperature in Kelvin, pressure in Pascals, and volume in cubic meters), \( R \) helps predict how a gas behaves under different conditions. Remember, correctly applying the gas constant in your calculations is vital for accurate results.

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Most popular questions from this chapter

Insects do not have lungs as we do, nor do they breathe through their mouths. Instead, they have a system of tiny tubes, called tracheae, through which oxygen diffuses into their bodies. The tracheae begin at the surface of an insect鈥檚 body and penetrate into the interior. Suppose that a trachea is 1.9 \(\mathrm{mm}\) long with a cross- sectional area of \(2.1 \times 10^{-9} \mathrm{m}^{2} .\) The concentration of oxygen in the air outside the insect is \(0.28 \mathrm{kg} / \mathrm{m}^{3},\) and the diffusion constant is \(1.1 \times 10^{-5} \mathrm{m}^{2} / \mathrm{s} .\) If the mass per second of oxygen diffusing through \(\mathrm{a}\) trachea is \(1.7 \times 10^{-12} \mathrm{kg} / \mathrm{s}\) , find the oxygen concentration at the interior end of the tube.

ssm A cylindrical glass beaker of height 1.520 \(\mathrm{m}\) rests on a table. The bottom half of the beaker is filled with a gas, and the top half is filled with liquid mercury that is exposed to the atmosphere. The gas and mercury do not mix because they are separated by a frictionless movable piston of negligible mass and thickness. The initial temperature is 273 \(\mathrm{K}\) . The temperature is increased until a value is reached when one-half of the mercury has spilled out. Ignore the thermal expansion of the glass and the mercury, and find this temperature.

A container holds 2.0 \(\mathrm{mol}\) of gas. The total average kinetic energy of the gas molecules in the container is equal to the kinetic energy of an \(8.0 \times 10^{-3} \mathrm{-kg}\) bullet with a speed of 770 \(\mathrm{m} / \mathrm{s}\) . What is the Kelvin temperature of the gas?

Two ideal gases have the same mass density and the same absolute pressure. One of the gases is helium (He), and its temperature is 175 \(\mathrm{K}\) . The other gas is neon (Ne). What is the temperature of the neon?

The mass of a hot-air balloon and its occupants is 320 \(\mathrm{kg}\) (excluding the hot air inside the balloon). The air outside the balloon has a pressure of \(1.01 \times 10^{5}\) Pa and a density of 1.29 \(\mathrm{kg} / \mathrm{m}^{3} .\) To lift off off, the air inside the balloon is heated. The volume of the heated balloon is 650 \(\mathrm{m}^{3} .\) The pressure of the heated air remains the same as the pres- sure of the outside air. To what temperature (in kelvins) must the air be heated so that the balloon just lifts off? The molecular mass of air is 29 u.
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