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Chapter 10: Problem 29

(a) Write the ideal-gas equation, and give the units used for each term in the equation when \(R=0.0821 \mathrm{~L}-\mathrm{atm} / \mathrm{mol}-\mathrm{K}\). (b) What is an ideal gas?

Short Answer

Expert verified
(a) The ideal gas equation is given by: \(PV = nRT\), with units P (atm), V (L), n (mol), R (0.0821 L路atm / mol路K), and T (K). (b) An ideal gas is a hypothetical gaseous substance whose behavior is independent of attractive and repulsive forces and can be described by the Ideal Gas Equation. It is a theoretical concept where gas particles have no volume and no interactions between them, and their behaviors are only determined by pressure, volume, and temperature.

Step by step solution

01

(a) Ideal gas equation and units

The ideal gas equation is given by: \[ PV = nRT \] Where: P = Pressure V = Volume n = Number of moles of the gas R = Ideal gas constant T = Temperature in Kelvin Given that R = 0.0821 L路atm / mol路K, the units for each term in the equation are as follows: P: Pressure is measured in atmospheres (atm) V: Volume is measured in Liters (L) n: Number of moles is measured in moles (mol) R: Ideal gas constant is in L路atm / mol路K T: Temperature is measured in Kelvin (K)
02

(b) Ideal gas definition

An ideal gas is a hypothetical gaseous substance whose behavior is independent of attractive and repulsive forces and can be described by the Ideal Gas Equation. In simpler terms, an ideal gas is a theoretical concept where gas particles have no volume and no interactions between them, and their behaviors are only determined by pressure, volume, and temperature. No real gas perfectly follows the ideal gas behavior, but many real gases exhibit behavior that closely approximates ideal gas behavior at high temperatures and low pressures.

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Key Concepts

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

Gas Behavior
Gas behavior refers to how gases respond to changes in pressure, volume, and temperature. The ideal gas law, expressed as \(PV = nRT\), provides a simplified model of this behavior. Here, \(P\) denotes pressure, \(V\) indicates volume, \(n\) is the number of moles, \(R\) is the gas constant, and \(T\) represents the temperature in Kelvin.

This equation assumes that gas particles are in constant, random motion and do not exert forces upon each other. In reality, gases deviate from ideal behavior due to intermolecular forces and the actual volume of gas particles. While no gas is truly ideal, many gases mimic ideal behavior, especially under conditions of high temperature and low pressure. These conditions reduce the effects of intermolecular forces, enabling gases to behave almost ideally.
Thermodynamics
Thermodynamics is a branch of physics that deals with the relationships between heat, work, and energy. It plays a vital role in understanding how gases behave under different conditions.

The ideal gas law is inherently linked to thermodynamics. It provides insights into how changes in a system's energy can influence the state of a gas. For example, if you increase the temperature of a gas while maintaining constant volume, the pressure increases due to the greater kinetic energy of gas molecules. This type of analysis helps in predicting the behavior of gases in processes such as combustion engines or atmospheric changes.

Understanding the fundamental principles of thermodynamics can simplify grasping complex gas behaviors and their applications in various scientific and engineering fields.
Gas Constant
The gas constant \(R\) is a crucial part of the ideal gas law and links various physical properties of gases under standardized conditions. It has the value \(R = 0.0821\, \ ext{L路atm/mol路K}\) used in the context of the ideal gas law.

The gas constant \(R\) ensures that the units in the ideal gas equation are consistent, allowing calculations of pressure, volume, temperature, or the number of moles to determine one unknown if the others are provided.
  • \(R\) helps to scale thermodynamic equations across different units and systems.
  • It represents the relationship between energy per degree and mole.

Understanding \(R\) enables one to solve problems involving gas behaviors, offering insight into how different gases will perform under a set of conditions. Moreover, applying \(R\) effectively requires consideration of the application of energy concepts in both ideal and real scenarios.

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Most popular questions from this chapter

A gas of unknown molecular mass was allowed to effuse through a small opening under constant-pressure conditions. It required \(105 \mathrm{~s}\) for \(1.0 \mathrm{~L}\) of the gas to effuse. Under identical experimental conditions it required \(31 \mathrm{~s}\) for \(1.0 \mathrm{Lof} \mathrm{O}_{2}\) gas to effuse. Calculate the molar mass of the unknown gas. (Remember that the faster the rate of effusion, the shorter the time required for effusion of \(1.0 \mathrm{~L} ;\) that is, rate and time are inversely proportional.)

Arsenic(III) sulfide sublimes readily, even below its melting point of \(320^{\circ} \mathrm{C}\). The molecules of the vapor phase are found to effuse through a tiny hole at \(0.28\) times the rate of effusion of Ar atoms under the same conditions of temperature and pressure. What is the molecular formula of arsenic(III) sulfide in the gas phase?

A gaseous mixture of \(\mathrm{O}_{2}\) and \(\mathrm{Kr}\) has a density of \(1.104 \mathrm{~g} / \mathrm{L}\) at 435 torr and \(300 \mathrm{~K}\). What is the mole percent \(\mathrm{O}_{2}\) in the mixture?

Nickel carbonyl, \(\mathrm{Ni}(\mathrm{CO})_{4}\), is one of the most toxic substances known. The present maximum allowable concentration in laboratory air during an 8 -hr workday is 1 part in \(10^{9}\) parts by volume, which means that there is one mole of \(\mathrm{Ni}(\mathrm{CO})_{4}\) for every \(10^{9}\) moles of gas. Assume \(24{ }^{\circ} \mathrm{C}\) and \(1.00 \mathrm{~atm}\) pressure. What mass of \(\mathrm{Ni}(\mathrm{CO})_{4}\) is allowable in a laboratory that is \(54 \mathrm{~m}^{2}\) in area, with a ceiling height of \(3.1 \mathrm{~m}\) ?

The Goodyear blimps, which frequently fly over sporting events, hold approximately \(175,000 \mathrm{ft}^{3}\) of helium. If the gas is at \(23^{\circ} \mathrm{C}\) and \(1.0 \mathrm{~atm}\), what mass of helium is in the blimp?
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