Problem 24 The universal gas constant \(R_{... [FREE SOLUTION] (2024)

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Chapter 1: Problem 24

The universal gas constant $R_{0}$ is equal to $49,700 \mathrm{ft}^{2}/\left(\mathrm{s}^{2} \cdot^{\circ} \mathrm{R}\right)$ or $8310 \mathrm{m}^{2}/\left(\mathrm{s}^{2} \cdot \mathrm{K}\right) .$ Show that these twomagnitudes are equal.

Short Answer

Expert verified

After converting the units and simplifying, we find that both magnitudes are equal. Thus, $49,700 \mathrm{ft}^{2} / \mathrm{s}^{2} °\mathrm{R} = 8310 \mathrm{m}^{2} / \mathrm{s}^{2}K$

Step by step solution

Conversion of feet to meters

Firstly, convert feet to meters: Knowing that 1ft = 0.3048m, the conversion can be expressed mathematically as $49,700 (ft)^{2} / s^{2}°R = 49,700 (0.3048m/1ft)^{2} / s^{2}°R$

Conversion of Fahrenheit to Kelvin

Secondly, convert Fahrenheit to Kelvin: The conversion factor between Fahrenheit and Kelvin is $K = (°F - 32) × 5/9 + 273.15$. Thus the Rankine temperature scale (°R) used in the English system, where 0 °R = absolute zero and uses Fahrenheit-degree intervals, can be converted to Kelvin (K) in the metric system by knowing that 1 °R = 0.55556 K. Therefore, the conversion can be expressed as $49,700 (0.3048m/1ft)^{2} / s^{2}°R = 49,700 (0.3048m/1ft)^{2} / s^{2}(1K / 0.55556°R)$

Evaluation of the expression

Simplify the above expression to obtain the resulting magnitude in Metric units.

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$Problem 24 The universal gas constant \(R_{... [FREE SOLUTION] (3)$

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