Calculate the universal gas constant for one gram molecule of gas. (a) 8.31 J/mol-K (b) 83.1 J/mol-K (c) 8.31 x10-7 J/mol-K (d) 83.1 x10-7 J/mol-K

To calculate the universal gas constant for one gram molecule of gas, we need to understand what the universal gas constant (R) represents and how it is typically expressed. The universal gas constant is a fundamental constant in the ideal gas law, which is given by the equation PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is temperature in Kelvin.

Understanding the Universal Gas Constant

The universal gas constant is usually expressed in terms of energy per temperature increment per mole. The most common value for R is 8.31 J/(mol·K), which is applicable when dealing with the ideal gas law in SI units.

Converting to One Gram Molecule

A gram molecule of gas refers to the amount of substance that has a mass in grams equal to its molecular weight. For example, one gram molecule of oxygen (O₂) has a mass of approximately 32 grams. However, when we talk about the universal gas constant, we often refer to it in terms of moles. Since one mole of any substance contains Avogadro's number of molecules (approximately 6.022 x 10²³), the gas constant remains the same regardless of the substance being considered.

Calculating R for One Gram Molecule

When we consider the universal gas constant for one gram molecule, we can use the value of R directly. Since 1 mole is equivalent to 1 gram molecule for gases, the universal gas constant remains 8.31 J/(mol·K) for one gram molecule of gas.

Evaluating the Options

Now, let’s look at the options provided:

• (a) 8.31 J/mol-K
• (b) 83.1 J/mol-K
• (c) 8.31 x 10^-7 J/mol-K
• (d) 83.1 x 10^-7 J/mol-K

From our understanding, the correct value for the universal gas constant is 8.31 J/(mol·K), which corresponds to option (a). The other options either scale the value incorrectly or introduce unnecessary factors that do not apply to the universal gas constant.

Final Thoughts

In summary, the universal gas constant for one gram molecule of gas is indeed 8.31 J/(mol·K). This value is crucial for calculations involving gases and is a fundamental part of thermodynamics and physical chemistry. If you have any further questions about gas laws or related topics, feel free to ask!