Problem 130 - General Chemistry: Principles and Modern Applications

Question

In your own words, define or explain each term or symbol. (a) atm; (b) STP; (c) \(R ;\) (d) partial pressure; (e) \(u_{\mathrm{rms}}\).

Step-by-Step Solution

Answer

(a) atm is a unit of pressure representing the atmospheric pressure at sea level. (b) STP stands for Standard Temperature and Pressure, at which the temperature is 0 degrees Celsius and the pressure is 1 atmosphere. (c) \(R\) is the ideal or molar gas constant. (d) Partial pressure is the pressure one gas in a mixture would exert if it were alone in the same volume at the same temperature. (e) \(u_{\mathrm{rms}}\) is the root mean square speed of particles in a gas.

Step 1: Define atm

The term 'atm' stands for atmosphere, which is a unit of pressure. It is defined as the pressure exerted by the weight of the atmosphere at sea level and is approximately 101325 pascals.

Step 2: Explain STP

STP stands for Standard Temperature and Pressure, a standard reference condition in the study of gases. At STP, temperature is 0 degrees Celsius (273.15 kelvin) and pressure is 1 atmosphere.

Step 3: Define \(R\)

In the context of gas laws, \(R\) usually represents the ideal gas constant, also known as the molar gas constant. Its value is approximately 8.314 J/(mol·K).

Step 4: Define partial pressure

Partial pressure is the pressure that a gas in a mixture of gases would exert if it occupied the same volume alone at the same temperature. It's a component of Dalton's Law of partial pressures.

Step 5: Explain \(u_{\mathrm{rms}}\)

The term \(u_{\mathrm{rms}}\) represents the root mean square speed of the particles in a gas. It is calculated by taking the square root of the average of the squares of the speeds of the particles in the gas.

Key Concepts

Pressure Units

Pressure is commonly expressed in different units, one of which is the "atm" or atmosphere. It serves as a standard measure of pressure, particularly in environmental and scientific contexts. At sea level, the atmospheric pressure is about 1 atm, equivalent to 101,325 pascals (Pa). This unit helps in comparing pressures in different scenarios, such as in weather systems or chemical reactions.

Other units of pressure you might encounter include pounds per square inch (psi), bar, and millimeters of mercury (mmHg). Understanding these different units and their conversions is essential, as it allows scientists and engineers to communicate data clearly across various disciplines.

• 1 atm = 101,325 Pa
• 1 atm = 760 mmHg
• 1 atm ≈ 14.696 psi

Knowing these conversions helps you switch between units effortlessly, a key skill when working with gas laws.

Standard Temperature and Pressure (STP)

When discussing gases, the term "Standard Temperature and Pressure" (STP) provides a universal reference. At STP, temperature is defined as 0 degrees Celsius, which is equivalent to 273.15 Kelvin. The pressure, meanwhile, is set at 1 atm.

STP is crucial for gas law calculations because it establishes a baseline for comparing different gases and their behaviors. By standardizing these conditions, scientists can predict how gases will react under certain circumstances, enabling them to design experiments and solve problems with greater accuracy.

Ideal Gas Constant

The ideal gas constant, denoted by the symbol "R," is a key part of the ideal gas law: \[ PV = nRT \]Here, \( P \) represents pressure, \( V \) stands for volume, \( n \) is the number of moles of the gas, and \( T \) is the absolute temperature in Kelvin.

"R" is approximately 8.314 J/(mol·K), serving as a bridge between macroscopic measurements (like volume and pressure) and the amount of substance and temperature. Understanding "R" and how it relates to the other variables in the ideal gas equation allows us to predict the behavior of ideal gases under various conditions.

• R = 8.314 J/(mol·K)
• Used in PV = nRT
• Links pressure, volume, temperature, and moles

Partial Pressure

In a mixture of gases, the concept of partial pressure explains how each gas contributes to the total pressure. Each gas in the mixture exerts pressure independently as if it were the only gas present. This is known as partial pressure.

Dalton's Law of Partial Pressures states that the total pressure of a gas mixture is the sum of the partial pressures of each individual gas. This principle is important in fields like chemistry and biology, where gases often interact in closed systems. Understanding partial pressures helps you calculate the effects each gas will have in a given situation.

• Each gas's pressure in a mixture
• Key in Dalton's Law
• Important for chemical reactions

Root Mean Square Speed

The root mean square speed \( u_{\mathrm{rms}} \) of gas particles provides insight into their kinetic energy. It is determined by taking the square root of the average of the squares of the speeds of each particle. This speed is directly related to the temperature and molar mass of the gas.

The formula for root mean square speed is:\[ u_{\mathrm{rms}} = \sqrt{\frac{3RT}{M}} \]Where \( R \) is the ideal gas constant, \( T \) is the temperature in Kelvin, and \( M \) is the molar mass of the gas. Understanding \( u_{\mathrm{rms}} \) helps explain how gas molecules behave at different temperatures, which is crucial in applications ranging from engine design to predicting atmospheric conditions.

• Relates to kinetic energy
• Depends on temperature and molar mass
• Helps in understanding gas behavior