A student proposed the following hypothesis regarding planetary atmospheres: *Hypothesis*:… — ACT

A student proposed the following hypothesis regarding planetary atmospheres:

*Hypothesis*: The rate of atmospheric escape of a gas from a planet's atmosphere is determined solely by the planet's surface temperature, such that planets with higher surface temperatures will experience higher rates of escape for all gases, regardless of the gravity of the planet or the molar mass of the gas.

To test this hypothesis, scientists measured the escape rates of helium ( $He$ , molar mass $\approx 4\text{ g/mol}$ ) and xenon ( $Xe$ , molar mass $\approx 131\text{ g/mol}$ ) from three different planets. The surface temperature, surface gravity (measured in units of $g$ , where $1\text{ }g = 9.8\text{ m/s}^2$ ), and escape rates of the gases are shown in the table below:

Planet	Surface Temperature ( $K$ )	Surface Gravity ( $g$ )	Helium ( $He$ ) Escape Rate (kg/s)	Xenon ( $Xe$ ) Escape Rate (kg/s)
Planet X	$300$	$0.40$	$1.2 \times 10^3$	$0.0$
Planet Y	$500$	$0.90$	$4.5 \times 10^2$	$0.0$
Planet Z	$500$	$0.40$	$8.8 \times 10^4$	$1.1 \times 10^1$

Based on these results, which of the following modifications to the student's hypothesis is best supported by the data?

A
The escape rate of a gas is determined solely by the planet's surface gravity, because planets with higher surface gravity always experience higher escape rates of all gases.
B
The hypothesis should be retained without modification because Planet Z has the highest surface temperature and also exhibits the highest escape rates for both gases.
The escape rate of a gas depends on both the planet's surface gravity and the molar mass of the gas, with lower surface gravity and lower molar mass leading to higher escape rates.Answer
D
The escape rate of a gas is inversely proportional to temperature and directly proportional to molar mass, meaning heavier gases escape faster from colder planets.

Answer

The escape rate of a gas depends on both the planet's surface gravity and the molar mass of the gas, with lower surface gravity and lower molar mass leading to higher escape rates.

The correct option correctly states that the escape rate depends on both surface gravity and molar mass, with lower gravity and lower molar mass promoting higher escape rates. This is supported by comparing Planet X and Planet Y, where the higher gravity of Planet Y results in a lower helium escape rate despite its higher temperature, and by comparing the escape rates of Helium and Xenon on Planet Z, where the lighter gas (Helium) escapes at a much faster rate.

Step-by-Step Solution

Analyze the student's initial hypothesis.

The hypothesis claims that atmospheric escape rate depends solely on surface temperature, and is independent of gravity or molar mass.

To evaluate or modify a hypothesis, we must first understand its core assumptions.

Compare the Helium escape rates of Planet X and Planet Y to test if temperature is the sole factor.

Planet Y has a higher temperature (

500\text{ K}

) than Planet X (

300\text{ K}

), but a lower Helium escape rate (

4.5 \times 10^2\text{ kg/s}

vs.

1.2 \times 10^3\text{ kg/s}

). This difference corresponds to Planet Y's higher surface gravity (

0.90\text{ }g

vs.

0.40\text{ }g

This demonstrates that surface gravity is a confounding factor that must be included in the hypothesis, and that higher gravity decreases the escape rate.

Compare the escape rates of Helium and Xenon on Planet Z to evaluate the effect of molar mass.

On Planet Z, Helium (molar mass

\approx 4\text{ g/mol}

) has a much higher escape rate (

8.8 \times 10^4\text{ kg/s}

) than Xenon (molar mass

\approx 131\text{ g/mol}

, escape rate

1.1 \times 10^1\text{ kg/s}

This shows that the molar mass of the gas also affects the escape rate, with lighter gases escaping more readily, which invalidates the claim that escape rate is independent of molar mass.

Synthesize these findings to identify the correct modification.

The hypothesis must be modified to include both surface gravity (inversely related to escape rate) and gas molar mass (inversely related to escape rate).

This directly addresses all the contradictions found between the original hypothesis and the experimental data.

Key Concept

Evaluating and modifying a hypothesis based on multi-variable experimental data.

Estimated Time:2m 30s

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