Question

Containers A and B have same gases. Pressure, volume and temperature of \(\mathrm{A}\) are all twice that of \(\mathrm{B}\), then the ratio of number of molecules of \(\mathrm{A}\) and \(\mathrm{B}\) are (a) \(1: 2\) (b) 2 (c) \(1: 4\) (d) 4

Short answer

The ratio of the number of molecules in A to B is 2.

Step-by-step solution

Understand the Relationship

According to the ideal gas law, the 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 temperature. In this problem, all conditions of container \( A \) are twice that of container \( B \). This means \( P_A = 2P_B, \), \( V_A = 2V_B, \) and \( T_A = 2T_B \).

Express Conditions for Container B

For container \( B \), the ideal gas equation is \( P_B V_B = n_B R T_B \). Here, \( n_B \) represents the number of moles in container \( B \).

Express Conditions for Container A

For container \( A \), substitute the doubled conditions into the ideal gas equation: \( 2P_B \cdot 2V_B = n_A R \cdot 2T_B \). Simplify to get \( 4P_B V_B = 2n_A R T_B \).

Solve for Number of Moles in A

Simplify the equation from Step 3: \( 4P_B V_B = 2n_A R T_B \) simplifies to \( 2P_B V_B = n_A R T_B \).

Ratio of Moles in Container A and B

We have the equation for container \( B \) as \( P_B V_B = n_B R T_B \). Comparing with \( 2P_B V_B = n_A R T_B \), divide both sides of the equation from Step 2 by the equation from Step 4 to find the ratio: \( \frac{n_A}{n_B} = \frac{2P_B V_B / R T_B}{P_B V_B / R T_B} = 2 \).

Conclusion on Number of Molecules

The number of molecules is proportional to the number of moles (since \( n = \frac{N}{N_A} \), where \( N \) is the number of molecules and \( N_A \) is Avogadro's number), therefore the ratio of the number of molecules \( N_A \) to \( N_B \) is the same as \( \frac{n_A}{n_B} \), which is 2.

Key concepts

Pressure

Pressure is one of the fundamental concepts when dealing with gases. It is defined as the force exerted by gas molecules colliding with the walls of their container. The unit for measuring pressure is the Pascal (Pa) or atmospheres (atm), depending on the context. These gas molecules move rapidly in random directions, causing impacts. - In the example of containers A and B, the pressure of container A is said to be twice as high as that of container B. - According to the ideal gas law, pressure is directly proportional to both the number of gas molecules and the temperature. Understanding pressure is crucial because it affects how gases behave under different conditions. When you increase the pressure by compressing the volume or increasing the temperature, you create a more crowded environment for the gas molecules, which in turn affects how they interact and occupy the space.

Volume

Volume is the space that a gas occupies. Like pressure, volume is a fundamental component of the ideal gas law equation, which can be described as \( PV = nRT \). In cases like our containers A and B, where the volume of A is twice that of B, it highlights the relationship between volume and the other factors in the equation.- Volume is measured in liters (L) or cubic meters (m³).- It is inversely related to pressure, as illustrated by Boyle’s Law, where increasing the volume decreases the pressure if temperature and moles remain constant.In the context of container A, having a volume twice as big as container B implies that, under the same conditions of moles and temperature, A will exert less pressure on its walls if the other factors aren't doubled accordingly. This becomes crucial when evaluating the behavior of gases under transformation, as in exercises involving ideal gas law calculations.

Temperature

Temperature is a measure of the average kinetic energy of gas particles. In the context of the ideal gas law, it is expressed in Kelvin (K) rather than Celsius or Fahrenheit to maintain proportionality in calculations. An understanding of temperature is critical because it directly influences gas behavior. - Temperature in container A is twice that of container B in this exercise. - Higher temperatures increase the energy of the particles, causing them to move more rapidly and collide with the container walls with greater force, thus increasing the pressure. Temperature affects not only pressure but also volume and the number of moles if they are allowed to vary. The linear relationship is captured in the equation form of Charles's Law, which states that volume is directly proportional to temperature when pressure is constant.

Moles

Moles represent the quantity of a gas, providing a link between macroscopic observations and microscopic properties. The mole is an SI unit counting the number of particles, such as atoms or molecules, in a given sample. One mole corresponds to Avogadro's number, approximately \(6.022 \times 10^{23}\) entities.- In the given problem, the number of moles is used to find the ratio of gas molecules between two containers, A and B.- The calculation of moles uses the ideal gas law. Since the conditions (pressure, volume, and temperature) of container A are double those of container B, the ratio of the number of moles is deduced to be 2.Understanding moles allows us to quantify how much of a substance is present in terms of its fundamental particles. It’s key to balancing chemical reactions and predicting the movement and interaction of gases as described in the kinetic molecular theory.