Question

A vessel contains 1 mole of \(O_{2}\) gas (relative molar mass 32 ) at a temperature \(\mathrm{T}\). The pressure of the gas is \(\mathrm{P}\). An identical vessel containing 1 mole of He gas (relative molar mass 4 ) at a temperature \(2 \mathrm{~T}\) has pressure of \(.\) (A) \(8 \mathrm{P}\) (B) \((\mathrm{p} / 8)\) (C) \(2 \mathrm{P}\) (D) \(\mathrm{P}\)

Short answer

The pressure of the Helium gas in the vessel is (C) \(2 \mathrm{P}\).

Step-by-step solution

Define the Ideal Gas Law equation for both cases

First, we need to write down the Ideal Gas Law equation for both the Oxygen gas and Helium gas cases. For Oxygen gas (case 1), we have: \(P_1 V_1 = n_1 R T_1\) And for Helium gas (case 2), we have: \(P_2 V_2 = n_2 R T_2\)

Use given information to simplify and solve

We are given the following information: 1. \(n_1 = 1\) mole of Oxygen gas 2. \(n_2 = 1\) mole of Helium gas 3. \(molar~mass_1 = 32\) and \(molar~mass_2 = 4\) 4. \(T_1 = T\) for Oxygen and \(T_2 = 2T\) for Helium 5. Both vessels are identical, so \(V_1 = V_2\) Using this information, we can rewrite the two equations as follows: \(P_1 V = R T\) \(P_2 V = R (2T)\)

Find the relation between the pressures

Divide the equation for Helium gas by the equation for Oxygen gas: \(\frac{P_2 V}{P_1 V}=\frac{R(2T)}{RT}\) Since V and R are the same for both cases, they cancel out: \(\frac{P_2}{P_1} = \frac{2T}{T}\) Thus, we find the relation between the pressures of the two gases: \(P_2 = 2P_1\)

Match the relation with the given options

We now have found the relation between the pressures of the two gases, which is \(P_2 = 2P_1\). To find the correct answer, we need to match this relation with the given options: (A) \(8 \mathrm{P}\) -> This is not correct, as it implies \(P_2 = 8P_1\) (B) \((\mathrm{p} / 8)\) -> This is not correct, as it implies \(P_2 = \frac{P_1}{8}\) (C) \(2 \mathrm{P}\) -> This is the correct option, as it implies \(P_2 = 2P_1\) (D) \(\mathrm{P}\) -> This is not correct, as it implies \(P_2 = P_1\) So the correct option is (C) \(2 \mathrm{P}\).

Key concepts

Molar Mass

Molar mass is a key concept in chemistry. It helps us understand how much a single mole of a substance weighs. This weight is measured in grams per mole (g/mol). A mole is a unit that counts particles like atoms or molecules. For example, Oxygen \(O_2\) has a molar mass of 32 g/mol. This value means that one mole of \(O_2\) weighs 32 grams.
In contrast, Helium, a lighter element, has a molar mass of only 4 g/mol, making it significantly less dense than oxygen. Understanding molar mass is important in calculations because it helps compare different gases. It also tells us about the volume behavior of the gas under the same conditions. The molar mass helps calculate the amount of substance (in moles) present, allowing us to use the Ideal Gas Law correctly.

Pressure

Pressure is the force applied by gas particles when they collide with the walls of their container. It is usually measured in units like pascals (Pa) or atmospheres (atm). Pressure is one of the critical variables in the Ideal Gas Law, which states the relationship between pressure, volume, temperature, and number of particles.
In our exercise, the pressure \(P\) for Oxygen at temperature \(T\) is compared with that of Helium at a higher temperature \(2T\). The fact that Helium exerts double the pressure of Oxygen under these conditions exemplifies how temperature impacts gaseous pressure when the number of moles and volume remains constant. The relationship between pressure and other gas properties is critical when predicting how gases behave under various physical conditions.

Temperature

Temperature measures how much energy particles in a system have. When the temperature increases, gas particles move faster. This increased movement makes gas particles collide with the container walls more often and with greater force, leading to higher pressure.
The exercise presents gases at different temperatures: Oxygen at \(T\) and Helium at \(2T\). Even though both gases have the same number of moles and volume, their pressures differ due to their temperature difference. In the Ideal Gas Law, temperature must always be in Kelvin, as it relates to the absolute energy gas particles have. Understanding how temperature affects gas behavior is vital for correctly applying and manipulating the Ideal Gas Law.

Gas Constant

The gas constant \(R\) is a universal constant in gas law equations, such as the Ideal Gas Law \(PV = nRT\). The value of \(R\) depends on the units used, but it remains the same across different gases. This constant helps express the relationship between pressure, volume, number of moles, and temperature for a gas.
The constant \(R\) bridges the molecular scale with the macroscopic scale, effectively linking particle behavior to observable properties like pressure or volume. In calculations, \(R\) ensures consistency within the Ideal Gas Law, allowing the direct comparison of different gases under equivalent conditions. This way, \(R\) is crucial for understanding the predictable behavior of ideal gases across various chemical processes.