Question

The relations between various variables of gaseous substances are given along with their formulae. Mark the incorrect relationship. (a) Density and molar mass : \(M=\frac{d R T}{P}\) (b) Universal gas constant, \(P, V, T: R=\frac{P V}{n T}\) (c) Volume and pressure: \(V_{2}=\frac{P_{2} V_{1}}{P_{1}}\) (d) Volume and temperature: \(V_{2}=\frac{V_{1} T_{2}}{T_{1}}\)

Short answer

Option (c) is incorrect.

Step-by-step solution

Reviewing Option (a)

The formula in option (a) is used to find the molar mass (M) of a gaseous substance. According to the ideal gas law, PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature. The density (d) is the mass (m) divided by volume (V), and m/n is the molar mass (M). Therefore, the formula becomes M = (m/V)RT/P or M = dRT/P which is correctly stated in option (a).

Reviewing Option (b)

Given the ideal gas equation PV = nRT, to find the universal gas constant R, we rearrange the equation as R = PV/nT. This is the correct relationship as given in option (b).

Reviewing Option (c)

Boyle's Law states that at a constant temperature, the volume of a gas is inversely proportional to the pressure. Mathematically, P1V1 = P2V2, but rearranging it as V2 = P1V1/P2 is incorrect since the relationship is inversely proportional. The correct formula should be V2 = V1P1/P2. Therefore, option (c) states the incorrect relationship.

Reviewing Option (d)

According to Charles's Law, at a constant pressure, the volume of a gas is directly proportional to its temperature in Kelvin. Hence, V1/T1 = V2/T2 or V2 = V1T2/T1, which shows that option (d) correctly represents this relationship.

Key concepts

Boyle's Law

Boyle's Law is a fundamental principle in the study of gases that describes the relationship between the pressure and volume of a gas at a constant temperature. Named after Robert Boyle, who first formulated the law in the 17th century, it states that the volume of a gas is inversely proportional to its pressure when temperature is held constant. This means if you increase the pressure exerted on a gas, its volume will decrease, provided the temperature doesn't change. The mathematical expression of Boyle's Law is given by the equation: \[ P_1V_1 = P_2V_2 \] To visualize this, imagine compressing a syringe with a fixed amount of gas inside; as the pressure increases due to the plunger being pushed down, the volume of gas inside the syringe decreases. Conversely, if you pull the plunger up, the pressure decreases and the volume of gas expands. This behavior is a cornerstone in understanding how gases respond to changing pressures and is essential in various applications, from breathing mechanisms in biology to the operation of pneumatic systems in industry.

When working with Boyle's Law in problems, accuracy in the manipulation of pressure and volume values is crucial. As seen in the exercise solution, a common mistake is to confuse the inversely proportional relationship with a direct proportion, leading to incorrect formulas and outcomes.

Charles's Law

Charles's Law, named after Jacques Charles, is another important gas law that explores the relationship between the volume and temperature of a gas. Stated simply, Charles's Law indicates that the volume of a gas is directly proportional to its absolute temperature, assuming the pressure and the amount of gas remain constant. This direct proportionality can be captured in the formula: \[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \] The key to applying Charles's Law is to always use temperatures in Kelvin, as this scale starts at absolute zero and ensures a straightforward proportionality. Imagine a balloon on a cold winter day—it shrinks because the colder temperature decreases the volume of the gas inside. If you take the same balloon into a warm room, it will expand as the gas's volume increases with the rising temperature. Understanding Charles's Law is invaluable when studying thermal processes, designing heating and cooling systems, and even predicting weather patterns, where the expansion and contraction of atmospheric gases play a critical role.

As demonstrated in the textbook solution, knowing how to apply Charles’s Law correctly can help students avoid mistakes in their calculations, particularly when converting temperatures to the Kelvin scale and when setting up proportional relationships between volume and temperature.

Gas Constant

The gas constant, often represented by the symbol R, is an essential element in the equations for Boyle's Law and Charles's Law when combined into the ideal gas law. It is termed 'universal' because it has the same value for all ideal gases, and it ties together the pressure, volume, temperature, and number of moles of gas in these equations. The value of R is derived from experimentally determined constants and has a value of approximately 8.314 J/(mol·K) in the International System of Units (SI). The gas constant appears in the ideal gas law, which is an equation of state for a hypothetical ideal gas and serves as a good approximation for real gases under a range of conditions. The ideal gas law is written as: \[ PV = nRT \] In this equation, P stands for pressure, V for volume, n for the number of moles of gas, and T for the absolute temperature. The gas constant connects these variables and allows for the calculation of one when the others are known. It's important to remember that R is a constant only when using standard units for pressure, volume, temperature, and moles. Using different units may require a different value of R.

Understanding the role of the gas constant is fundamental to solving problems in chemistry, physics, and engineering that involve gas behaviors. As the textbook solution points out, the correct use of R in gas law equations is critical to deriving accurate relationships between the physical properties of gases.