A student measured the speed of sound in a chamber filled with pure carbon dioxide (CO_2) … — ACT

A student measured the speed of sound in a chamber filled with pure carbon dioxide ( $CO_2$ ) gas at various temperatures. The measured speed of sound, in meters per second ( $\text{m/s}$ ), at each temperature, in degrees Celsius ( $^\circ\text{C}$ ), is shown in the table below:

Temperature ( $^\circ\text{C}$ )	Speed of Sound ( $\text{m/s}$ )
$0$	$259$
$20$	$268$
$40$	$277$
$60$	$286$

Assuming the speed of sound continues to change at a constant rate with respect to temperature, what is the predicted speed of sound in $CO_2$ gas, in meters per second ( $\text{m/s}$ ), at a temperature of $100^\circ\text{C}$ ?

Answer: 304 m/s

Answer

The predicted speed of sound in carbon dioxide gas at

100^\circ\text{C}

304\text{ m/s}

The speed of sound increases linearly by

9\text{ m/s}

for every

20^\circ\text{C}

increase in temperature, which is a rate of

0.45\text{ m/s}

per

1^\circ\text{C}

. The target temperature of

100^\circ\text{C}

40^\circ\text{C}

higher than the highest data point in the table (

60^\circ\text{C}

). The speed of sound will therefore increase by

40 \times 0.45 = 18\text{ m/s}

beyond the

60^\circ\text{C}

speed. Adding this to

286\text{ m/s}

yields

304\text{ m/s}

Step-by-Step Solution

Determine the constant rate of change of the speed of sound per

1^\circ\text{C}

temperature increase.

The speed of sound increases at a rate of

0.45\text{ m/s}

per

1^\circ\text{C}

This establishes the linear trend shown in the experimental data.

Find the temperature interval between the highest measured data point and the target temperature.

The difference is

40^\circ\text{C}

(from

60^\circ\text{C}

100^\circ\text{C}

This determines how far outside the measured data range the extrapolation must extend.

Multiply the temperature interval by the rate of change and add it to the speed of sound at the highest measured temperature.

286\text{ m/s} + (40^\circ\text{C} \times 0.45\text{ m/s/}^\circ\text{C}) = 304\text{ m/s}

This completes the linear extrapolation to predict the final value.

Key Concept

Linear extrapolation relies on determining a constant rate of change from the given data points and applying it to a target value outside the experimental range.

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