Question

If pressure and temperature of an ideal gas are doubled and volume is halved, the number of molecules of the gas
(A) Becomes half
(B) Becomes two times
(C) Becomes four times
(D) Remains constant

Answer 1

Hint Use the equation of ideal gas law. Use Boyle’s law equation to find the relation between pressure and volume. Apply the given conditions to the ideal gas law equation and find what happens to the n value.

Complete Step By Step Solution
We know the ideal gas law is obtained from Boyle's law. Boyle’s Law states that for a fixed mass of gas at a constant temperature, the volume of the gas is inversely proportional to the pressure of the gas.
Which means that
$$P\propto {\displaystyle \frac{1}{V}}$$
Where P is the pressure of the gas and V is the volume of the gas molecule
Now,
$$PV=C$$ , where C defines a constant
$$PV=nRT$$ , Where n is number of moles of gas molecules of fixed mass, R is gas constant and T is the temperature of the gas
Now in our case, it is given as Pressure of the ideal gas is doubled and Volume of the ideal gas is halved.
This means that
$$P=2P$$ and $$V={\displaystyle \frac{V}{2}}$$
Applying this condition to the ideal gas equation given above, we get
$$2P\times {\displaystyle \frac{V}{2}}=nRT$$
It is also given that the temperature of the gas is doubled. Applying this to the above mentioned equation, we get,
$$P\times V=nRT\times 2$$
Taking n on one side, we get
$$n={\displaystyle \frac{P\times V}{2\times RT}}$$
From the above equation it is seen that, number of molecules or moles is halved , when the pressure and temperature are increased and volume is halved.

Thus , Option(A) is the right answer.

Note
The given question can also be solved by considering Charles law, which states that for a fixed mass at constant pressure on the gas, the volume of the gas is directly proportional to the temperature of the gas.