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Chapter 5: Problem 64

What will be the effect on the volume of an ideal gas if the pressure is doubled and the absolute temperature is halved?

Short Answer

Expert verified
The volume of the ideal gas will decrease to one-fourth of its initial value when the pressure is doubled and the absolute temperature is halved.

Step by step solution

01

Remember 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, R is the ideal gas constant, and T is the absolute temperature.
02

Analyze the given conditions

We are given that the pressure is doubled and the absolute temperature is halved. Using P' to denote the new pressure and T' to denote the new temperature, we have: \(P' = 2P\) \(T' = \frac{1}{2}T\)
03

Determine the initial and final gas law equations

Using the initial pressure P, volume V, and temperature T, the initial gas law equation is: \(PV = nRT\) For the new pressure P' and temperature T', but the same amount of gas n, we have: \(P'V' = nR( T' )\)
04

Insert the new pressure and temperature values into the second equation

Replace P' with 2P and T' with (1/2)T in the second equation. This gives us: \((2P)V' = nR(\frac{1}{2}T)\)
05

Simplify the equation

Simplify the equation and solve for V': \(2PV' = \frac{1}{2} nRT\) Now notice that the term \(PV = nRT\), so we can replace \(nRT\) with \(PV\) in the equation above: \(2PV' = \frac{1}{2} PV\)
06

Solve for the new volume, V'

Divide both sides of the equation by 2P to find the new volume, V': \(V' = \frac{1}{4} V\)
07

Interpret the result

The new volume, V', is one-quarter of the initial volume, V. This means that if the pressure is doubled and the absolute temperature is halved, the volume of the ideal gas will decrease to one-fourth of its initial value.

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Most popular questions from this chapter

Hyperbaric oxygen therapy is used to treat patients with carbon monoxide poisoning as well as to treat divers with the bends. In hyperbaric oxygen therapy, a patient is placed inside a 7.0-ft cylinder with a 3.0-ft diameter, which is then filled with oxygen gas to a total pressure of 2.50 atm. Assuming the patient takes up 32.0% of the chamber’s volume, what volume of \(\mathrm{O}_{2}(g)\) from a gas cylinder at \(25^{\circ} \mathrm{C}\) and 95 \(\mathrm{atm}\) is required to fill the chamber to a total pressure of 2.50 \(\mathrm{atm}\) at \(25^{\circ} \mathrm{C} ?\) Assume the hyperbaric chamber initially contains air at 1.00 atm before \(\mathrm{O}_{2}(g)\) is added.

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