Question

Kinetic energy per mole of an ideal gas is (a) Zero at zero Kelvin temperature (b) Independent of temperature (c) Proportional to the absolute temperature of the gas (d) Proportional to pressure at constant temperature

Short answer

(c) Proportional to the absolute temperature of the gas

Step-by-step solution

Understanding Kinetic Energy in a Gas

The kinetic energy of one mole of an ideal gas can be determined using the equation for the average kinetic energy per molecule, which is given as \( \frac{3}{2} k_B T \), where \( k_B \) is the Boltzmann constant and \( T \) is the absolute temperature. To find the kinetic energy per mole, we multiply this by Avogadro's number \( N_A \).

Expression for Kinetic Energy Per Mole

The kinetic energy per mole of an ideal gas is formulated as \( \frac{3}{2} N_A k_B T = \frac{3}{2} R T \), where \( R \) is the universal gas constant. This derives from recognizing that \( N_A k_B \) equals \( R \).

Analyzing the Options

- Option (a): At zero Kelvin, the kinetic energy is zero implies that temperature factors directly, which aligns with the formula when \( T = 0 \).- Option (b): The formula \( \frac{3}{2} R T \) shows dependency on \( T \), thus, kinetic energy is not independent of temperature.- Option (c): The kinetic energy calculated is directly proportional to \( T \).- Option (d): Relates kinetic energy to pressure at constant \( T \), but at constant \( T \), energy derivatively remains linear to \( T \).

Conclusion on Correct Option

Considering the derivation and each option, the correct answer is option (c): The kinetic energy per mole of an ideal gas is proportional to the absolute temperature of the gas.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in physics and chemistry that describes the relationship between pressure, volume, and temperature of an ideal gas. When you think "ideal gas," imagine a simplified model of a gas where molecules do not interact except through elastic collisions. It's expressed as \( PV = nRT \), where:

• \( P \) is the pressure of the gas
• \( V \) is the volume it occupies
• \( n \) is the amount of substance (in moles)
• \( R \) is the universal gas constant
• \( T \) is the absolute temperature

This formula helps us understand how gases will behave under different conditions. Change any of the variables like temperature or volume, and you can predict how the gas's pressure will change. This equation is very useful when you need to calculate the effects of temperature changes on the properties of a gas. It ties in very closely with understanding kinetic energy, as temperature is a measure of the average kinetic energy of particles in a substance.

Boltzmann Constant

The Boltzmann constant \( k_B \) plays a key role in bridging macroscopic and microscopic physical quantities. It connects the average kinetic energy of particles with temperature. Its value, approximately \( 1.38 \times 10^{-23} \) Joules per Kelvin (J/K), is used to express energy per degree per particle.
In simpler terms, the Boltzmann constant is like a translator between the microscopic world of atoms and molecules and the macroscopic world we observe.
If you've seen the formula for the average kinetic energy of a molecule \( (\frac{3}{2} k_B T) \), that's where this constant comes in. Each particle's movement, described by this kinetic energy, links directly to the absolute temperature \( T \), with \( k_B \) as the factor that allows us to calculate this energy efficiently.
Using the Boltzmann constant helps us understand phenomena at the atomic level, translating small-scale movements into meaningful concepts like temperature.

Absolute Temperature

Absolute temperature, denoted in Kelvin (K), is a scale that begins at absolute zero, the point at which there is no thermal motion. Unlike other scales such as Celsius or Fahrenheit, the Kelvin scale is an absolute measure of temperature, providing a true zero point.
Absolute zero is considered the lowest limit of the thermodynamic temperature scale, equivalent to \(-273.15\) °C or \(-459.67\) °F, where molecular motion virtually stops.
Temperature in Kelvin simplifies many equations in physics by eliminating negative values, especially in gas laws and kinetic theory. When we look at the relationship \( rac{3}{2} R T \) for kinetic energy per mole, \( T \) must be in Kelvins to provide accurate results.
Absolute temperature is integral for accurately predicting how a gas will behave regarding its energy, with temperature acting as a direct measure of the system's energy state.

Universal Gas Constant

The universal gas constant \( R \) is a crucial component in the Ideal Gas Law and other equations in thermodynamics. Its value is approximately \( 8.314 \) Joules per mole per Kelvin (J/(mol·K)).
Think of \( R \) as the factor that scales the behavior of a single particle's energy to a mole's worth of particles. It shows up in expressions like \( PV = nRT \), linking energy with temperature and the number of particles.

• \( R \) facilitates calculations involving moles—one of chemistry's most important units—by scaling up properties from individual particles to quantities used in laboratory and industrial applications.
• In the context of kinetic energy \( rac{3}{2} RT \), the universal gas constant helps convert molecular insights into macroscopic understanding.

So, \( R \) is like a bridge, helping to understand energy principles on a larger scale, reflecting how a collection of molecules behaves in aggregate. This constant is important for both practical computing of gas behaviors and theoretical exploration of thermodynamics.