Question

When the gas constant \(R\) is given as \(8.31\) the units are (1) Lit atm mol 'deg ' (2) Cal mole ' deg \({ }^{1}\) (3) Joules mole ' deg \({ }^{1}\) (4) Ergs mole deg

Short answer

Option (3): Joules mole ' deg

Step-by-step solution

Understand the Gas Constant

The gas constant, denoted as \( R \), is a fundamental constant in chemistry and physics. It relates the energy scale of molecules to the temperature scale and is widely used in equations like the ideal gas law.

Identify Given Value and Units

In the problem, the gas constant \( R \) is given as \( 8.31 \). It's important to recall that \( R \) has specific units tied to energy, mole, and temperature.

Match \( R \) Value to Units

Given the value of \( R = 8.31 \), the most commonly associated units are Joules per mole per Kelvin. Check the provided options to find the matching units.

Compare Options

The provided options are: 1. Lit atm mol 'deg ' 2. Cal mole ' deg 3. Joules mole ' deg 4. Ergs mole deg Recognize that Joules are the standard in SI units for energy. Hence, \( 8.31 \text{ J mol}^{-1} \text{ K}^{-1} \) matches option (3).

Key concepts

ideal gas law

The ideal gas law is an equation of state for an ideal gas. It relates the pressure, volume, temperature, and the number of moles of gas to each other. The standard form of the ideal gas law is given by:
\( PV = nRT \)
Where:

• \( P \) is the pressure of the gas in pascals (Pa)
• \( V \) is the volume of the gas in cubic meters (m³)
• \( n \) is the number of moles of the gas
• \( R \) is the universal gas constant
• \( T \) is the temperature in kelvin (K)

The gas constant \( R \) plays a fundamental role in this equation. It's a bridge between macroscopic and microscopic properties of gases. In practical applications, the ideal gas law helps to understand and predict the behavior of gases under various conditions of temperature and pressure.

units of measurement

Units of measurement are crucial in scientific calculations as they provide a standard for reporting and interpreting data. The unit of the gas constant, \( R \), varies depending on the context but is often given as \( 8.314 \text{ J mol}^{-1} \text{ K}^{-1} \).
When working with the ideal gas law, make sure to use consistent units:

• Pressure \( P \) in pascals \( \text{Pa} \)
• Volume \( V \) in cubic meters \( \text{m}^3 \)
• Temperature \( T \) in kelvins \( \text{K} \)

In this problem, we noted that the gas constant was given as 8.31, which corresponds to the unit of Joules per mole per Kelvin \( \text{J mol}^{-1} \text{ K}^{-1} \). This is a typical unit for the gas constant in the International System of Units (SI). Being aware of such specific units helps in solving problems correctly and avoids confusion during calculations.

molecular energy

Molecular energy refers to the energy possessed by molecules due to their motion and interactions. The gas constant \( R \) is a crucial factor when studying molecular energies in gases.
One key aspect is how \( R \) links the average kinetic energy of gas molecules to temperature. According to kinetic molecular theory:
\( \frac{3}{2} k_B T = K_{avg} \)
where:

• \( k_B \) is the Boltzmann constant
• \( T \) is the absolute temperature
• \( K_{avg} \) is the average kinetic energy per molecule

Because \( R = N_A k_B \), where \( N_A \) is Avogadro's number, we can also write the expression for average kinetic energy per mole as:
\( \frac{3}{2} RT \)
This equation explains how the energy of gas molecules increases with temperature. Understanding this relationship is vital for comprehending behaviors of gases, such as why they expand when heated. Molecular energy is foundational to many fields, including thermodynamics and statistical mechanics.