A chemist adds a sample of an ideal gas to a balloon and measures its volume. The chemist removes some of the gas from the balloon and measures its volume again. If the chemist removes 21% of the gas from the balloon, and the pressure and temperature of the gas do not change, what happens to the volume of the balloon?

The volume of the gas decreases also in a 21% based on the Avogadro's law

Explanation:

Based on Avogadro's law, the volume of a gas is directely proportional to the amount of moles of gas when temperature and pressure remain constants.

The formula is:

V1 / n1 = V2 / n2

Where V is volume and n are moles in 1, initial state and 2, final state of the gas.

At beginning, V1 = 100%, n1 = 100%,

As the chemist removes 21% of the gas, 79% of moles remain = n2.

Replacing:

100% / 100% = V2 / 79%

V2 = 79%, that means:

part 1: the half-life of the element is 22 years.

part 2: it will take 132.02 year for the sample to decay from 308.0 g to 4.8125 g.

explanation:

part 1: there are two ways to determine the half life time.

first way: the half life time can be determined directly from its definition as it is the time required for the sample to decay to its half concentration.

from the data the initial concentration was 45.0 g and decayed to its half value (22.5 g) in 22.0 years.

second way: the radio active decay is considered first order reaction so we can use the laws of first order reactions.

k = (1/t) ln (a/a-x), where k is the rate constant of the reaction,

t is the time of the reaction

a is the initial concentration of the reactant

a-x is the remaining concentration at time (t)

a = 45.0 g the initial concentration at t = 0 year.

at t = 22 year, a-x = 22.5 g

then k = (1/22 year) ln (45.0 g/ 22.5 g) = 0.0315 year-1.

the half-life time (t1/2) = ln 2/ k = 0.693/ k =0.693/ 0.0315 = 22 years.

part 2:

using the same law k = (1/t) ln (a/a-x),

using the given data; a = 308.0 g and a-x = 4.8125 g and the calculated k from the part 1 (k = 0.0315 year-1)

t = (1/k) ln (a/a-x) = (1/0.0315) ln (308.0/ 4.8125) = 132.02 years