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Chapter 5: Problem 13

Explain how Boyle’s law, Charles’s law, and Avogadro’s law are special cases of the ideal gas law

Short Answer

Expert verified
Boyle's Law, Charles's Law, and Avogadro's Law are special cases of the ideal gas law, with the equation \( PV = nRT \). Boyle's Law is a special case when temperature and the number of moles are constant, so \( P_1V_1 = P_2V_2 \). Charles's Law is a special case when pressure and the number of moles are constant, resulting in \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \). Avogadro's Law is a special case when temperature and pressure are constant, giving \( \frac{V_1}{n_1} = \frac{V_2}{n_2} \).

Step by step solution

01

Write down the ideal gas law equation

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

Analyze Boyle's Law

Boyle's Law states that for a fixed amount of an ideal gas at constant temperature (i.e., when n and T are held constant), the pressure and volume of the gas are inversely proportional. Mathematically, this can be expressed as: \( P_1V_1 = P_2V_2 \)
03

Show Boyle's Law as a special case of the ideal gas law

Since n and T are constant in Boyle's Law, we can rewrite the ideal gas law equation as: \( PV = constant \) Now, let's substitute the initial pressure and volume as \( P_1 \) and \( V_1 \) and final pressure and volume as \( P_2 \) and \( V_2 \) in the ideal gas law equation: \( P_1V_1 = nRT \) and \( P_2V_2 = nRT \) Since both equations are equal to \( nRT \), we can equate the left-hand sides of the two equations: \( P_1V_1 = P_2V_2 \) Thus, Boyle's Law is a special case of the ideal gas law when the temperature and the number of moles are held constant.
04

Analyze Charles's Law

Charles's Law states that for a fixed amount of an ideal gas at constant pressure (i.e., when n and P are held constant), the volume and temperature of the gas are directly proportional. Mathematically, this can be expressed as: \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \)
05

Show Charles's Law as a special case of the ideal gas law

Since n and P are constant in Charles's Law, we can rewrite the ideal gas law equation as: \( V = \frac{nRT}{P} \) Now, let's substitute the initial volume and temperature as \( V_1 \) and \( T_1 \) and the final volume and temperature as \( V_2 \) and \( T_2 \) in the ideal gas law equation: \( V_1 = \frac{nRT_1}{P} \) and \( V_2 = \frac{nRT_2}{P} \) Dividing the first equation by the second equation gives: \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \) Thus, Charles's Law is a special case of the ideal gas law when the pressure and the number of moles are held constant.
06

Analyze Avogadro's Law

Avogadro's Law states that for a gas at constant temperature and pressure (i.e., when T and P are held constant), the volume of the gas is directly proportional to the number of moles. Mathematically, this can be expressed as: \( \frac{V_1}{n_1} = \frac{V_2}{n_2} \)
07

Show Avogadro's Law as a special case of the ideal gas law

Since T and P are constant in Avogadro's Law, we can rewrite the ideal gas law equation as: \( V = \frac{nRT}{P} \) Now, let's substitute the initial volume and moles as \( V_1 \) and \( n_1 \) and the final volume and moles as \( V_2 \) and \( n_2 \) in the ideal gas law equation: \( V_1 = \frac{n_1RT}{P} \) and \( V_2 = \frac{n_2RT}{P} \) Dividing the first equation by the second equation gives: \( \frac{V_1}{n_1} = \frac{V_2}{n_2} \) Thus, Avogadro's Law is a special case of the ideal gas law when the temperature and pressure are held constant.

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Most popular questions from this chapter

A compound containing only \(\mathrm{C}, \mathrm{H},\) and \(\mathrm{N}\) yields the following data: i. Complete combustion of 35.0 \(\mathrm{mg}\) of the compound produced 33.5 \(\mathrm{mg}\) of \(\mathrm{CO}_{2}\) and 41.1 \(\mathrm{mg}\) of $\mathrm{H}_{2} \mathrm{O} .$ ii. A 65.2 -mg sample of the compound was analyzed for nitrogen by the Dumas method (see Exercise \(137 ),\) giving 35.6 \(\mathrm{mL}\) of dry \(\mathrm{N}_{2}\) at \(740 .\) torr and \(25^{\circ} \mathrm{C}\) . iii. The effusion rate of the compound as a gas was measured and found to be \(24.6 \mathrm{mL} / \mathrm{min}\). The effusion rate of argon gas, under identical conditions, is \(24.6 \mathrm{mL} / \mathrm{min}\). What is the molecular formula of the compound?

Consider a sample of a hydrocarbon (a compound consisting of only carbon and hydrogen) at 0.959 atm and 298 \(\mathrm{K}\). Upon combusting the entire sample in oxygen, you collect a mixture of gaseous carbon dioxide and water vapor at 1.51 atm and 375 \(\mathrm{K}\). This mixture has a density of 1.391 g/L and occupies a volume four times as large as that of the pure hydrocarbon. Determine the molecular formula of the hydrocarbon

Without looking at a table of values, which of the following gases would you expect to have the largest value of the van der Waals constant $b : \mathrm{H}_{2}, \mathrm{N}_{2}, \mathrm{CH}_{4}, \mathrm{C}_{2} \mathrm{H}_{6},\( or \)\mathrm{C}_{3} \mathrm{H}_{8} ?$

A \(2.50-\mathrm{L}\) flask contains 0.60 \(\mathrm{g} \mathrm{O}_{2}\) at a temperature of \(22^{\circ} \mathrm{C}\) . What is the pressure (in atm) inside the flask?

Some very effective rocket fuels are composed of lightweight liquids. The fuel composed of dimethylhydrazine $\left[\left(\mathrm{CH}_{3}\right)_{2} \mathrm{N}_{2} \mathrm{H}_{2}\right]$ mixed with dinitrogen tetroxide was used to power the Lunar Lander in its missions to the moon. The two components react according to the following equation: $$\left(\mathrm{CH}_{3}\right)_{2} \mathrm{N}_{2} \mathrm{H}_{2}(l)+2 \mathrm{N}_{2} \mathrm{O}_{4}(l) \longrightarrow 3 \mathrm{N}_{2}(g)+4 \mathrm{H}_{2} \mathrm{O}(g)+2 \mathrm{CO}_{2}(g)$$ If 150 g dimethylhydrazine reacts with excess dinitrogen tetroxide and the product gases are collected at \(127^{\circ} \mathrm{C}\) in an evacuated 250-L tank, what is the partial pressure of nitrogen gas produced and what is the total pressure in the tank assuming the reaction has 100% yield?
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