The product of pressure P and volume V of an ideal gas is directly proportional to the product of the moles n and absolute temperature T of the gas, as shown below. What is the value of the gas constant R derived from the graph of the data?

Question

The product of pressure P and volume V of an ideal gas is directly proportional to the product of the moles n and absolute temperature T of the gas, as shown below.  What is the value of the gas constant R derived from the graph of the data?

Quizplus · Accepted Answer

11ecba2d_123d_1431_a3bf_eb02d53dbedf_MD0008_00 An ideal gas is a hypothetical gas with idealized characteristics used to make a simplified model to predict the behavior of real gases. According to this model, the average kinetic energy 11ecba2d_123d_1432_a3bf_f71514ad42ea_MD0008_00 of a single gas molecule is directly proportional to temperature via the Boltzmann constant (k = 1.381 × 10⁻²³ J/K). When the Boltzmann constant is expressed per mole of a gas, it is known as the gas constant (R = 8.31 J/mol∙K): 11ecba2d_123d_1433_a3bf_89490d110aef_MD0008_00 Because pressure correlates to energy per unit volume, the joules unit of R can be converted to units of pressure and volume. For example, expressing R in units of liters and atmospheres gives: 11ecba2d_123d_1434_a3bf_4f39e551b4bb_MD0008_00 As a result, the exact numerical value of R depends on the units used to measure it. Accordingly, when calculations with R are performed, the value of R used must match the units in the calculation. For example, the behavior of a gas can be described by the ideal gas law: 11ecba2d_123d_3b45_a3bf_79b300ea5d28_MD0008_00 where P is the pressure, V is the volume occupied, n is the number of moles, and T is the absolute temperature (in Kelvin). This relationship shows that PV is directly proportional to nT, and R is the proportionality constant: 11ecba2d_123d_3b46_a3bf_1d135e031a0b_MD0008_00 Therefore, a graph of PV (y-axis) vs. nT (x-axis) results in a straight line with a slope of R. Evaluating the slope of the line from the graph given in this question yields: 11ecba2d_123d_3b47_a3bf_0b2465f97f40_MD0008_00 11ecba2d_123d_3b48_a3bf_69e978859246_MD0008_00 (Choice A) 0.0160 mol∙K/mmHg∙m³ results from calculating the reciprocal slope (Δx/Δy), instead of the slope (Δy/Δx), of the line. (Choices B and C) Although 0.0821 L∙atm/mol∙K and 8.31 J/mol∙K are valid values of R expressed in alternate units, they are not the value of R derived from the graph expressed in the units of the given data. Educational objective: The average kinetic energy of a single ideal gas molecule is directly proportional to temperature via the Boltzmann constant, which can be expressed per mole of gas as the ideal gas constant R. According to the ideal gas law, PV is directly proportional to nT, and a graph of PV vs. nT results in a straight line with a slope of R.

The Product of Pressure P and Volume V of an Ideal