Question

Consider the following gaseous sample in a cylinder fitted with a movable piston. Initially there are n moles of the gas at temperature T, pressure P, and volume V. Choose the cylinder that correctly represents the gas after each of the following changes. (1) The pressure on the piston is tripled at constant n and T. (2) The temperature is doubled at constant n and P. (3) n moles of another gas are added at constant T and P. (4) T is halved and pressure on the piston is reduced to a quarter of its original value.

Answer 1

Answer:

The picture with the representations of the cylinders is attached.

The answers are:

• Question (1): Sample (b); volume is reduced by a factor of 1/3

• Question (2): sample (a); volume is doubled

• Question (3): Sample (c); volume is doubled

• Question (4): sample (a); volume is doubled.

Explanation:

Question (1) The pressure on the piston is tripled at constant n and T.

Sample b is occupying a much smaller volume than the original gas sample.

The condition of constant n and constant T were studied by the chemist scientist Robert Boyle. Constant n means that the amount (number of moles) of the gas has not changed. The results of his experiments and the results were summarized in the known Boyle's law.

Boyle's law state that at constant temperature, the volume and pressure of a fixed amount of gas are inversely related: PV = constant.

Then, since P has been tripled, the volume of the gas is reduced by a factor of 1/3, i.e. the final volume will be one third of the original volume, which is represented by the figure (b).

Question (2) The temperature is doubled at constant n and P.

When the temperatue is increased for a fixed amount of gas (constant n) at constant P, the gas will increase its volume. This is known as Charles' law: volume and temperature are directly related, V / T = constant.

Figure (a) represents a sample of gas where the space is more empty than in the original sample (the shaded region is clearer) but yet the volume has increased.

So, since V / T is constant, the doubling T means that V gets doubled too, which is shown by the sample (a).

Question (3) n moles of another gas are added at constant T and P.

The shaded region inside sample c shows a similar color than the original sample, which may be assumed as that, in spite of the volume has increased,  the "concentration" of gas particles has benn kept, meaning that the number of particles has also increased.

So, figure c represents an increase in volume due to the increase of the number of partilces.

That situation was studied by the scientist Avogadro. Avogadro's principle states that, for the same temperature and pressure, the volume occupied by a gas is proportional to the amount of gas measured as number of moles.

Hence, since the same number n of moles of gas has been added, the number of moles has been doubled, at constant T and P, causing that the volume gets doubled. That situation is represented by the sample c.

Question  (4) T is halved and pressure on the piston is reduced to a quarter of its original value.

What happens to a fixed amount of gas when both temperature and pressure are changed is determined by the combination of the laws: PV/T = constant.

So, the joint effect of halving T and reducing P to one quarter of its original value, for a fixed amount of gas (Avogadro's principle), can be seen by the relation:

• P₁V₁ /T₁ = P₂V₂/T₂

With T₂ = T₁ / 2 and P₂ = P₁ / 4

• P₁V₁ / T₁ = (P₁ / 4) V₂ / (T₁ / 2)

• P₁V₁ / T₁ = (2 / 4) × P₁V₂ / T₂

• V₁ = (1/2) V₂ ⇒ V₂ = 2 V₁

Hence, the final volume has been doubled, with the same number of particles (n); so it is also the figure (a).