Question

1. a. the mean molar mass of the atmosphere at the surface of the earth is \\(29.0\text{ g/mol}\\). earths surface temperature is \\(298\text{ k}\\) and its pressure is \\(1.00\text{ atm}\\). assuming ideal behavior, calculate the density of earths atmosphere under these conditions. show your work. box your final answer.

b. what assumptions have to be made in order to assume ideal behavior? how would the calculated density differ if the conditions were not ideal?

Explanation:

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<reasoning>

Identify given values and target variables

Using the Molar Mass of Gases and Ideal Gas Law Calculations knowledge points
\[

$$\begin{aligned} &M = 29.0\text{ g/mol}\\ &T = 298\text{ K}\\ &P = 1.00\text{ atm}\\ &R = 0.08206\text{ L}\cdot\text{atm}/(\text{mol}\cdot\text{K})\\ &\text{Target: Density } d = \frac{m}{V} \end{aligned}$$

\]

Derive the gas density formula

Using the Ideal Gas Law Calculations knowledge point
\[

$$\begin{aligned} &PV = nRT \implies PV = \frac{m}{M}RT\\ &d = \frac{m}{V} = \frac{PM}{RT} \end{aligned}$$

\]

Calculate the density of the atmosphere

Using the Ideal Gas Law Calculations knowledge point
\[

$$\begin{aligned} &d = \frac{(1.00\text{ atm})(29.0\text{ g/mol})}{(0.08206\text{ L}\cdot\text{atm}/(\text{mol}\cdot\text{K}))(298\text{ K})}\\ &d \approx 1.1859\text{ g/L} \approx 1.19\text{ g/L} \end{aligned}$$

\]

Analyze assumptions for ideal behavior

To assume ideal behavior, we assume:

1. The volume of individual gas particles is negligible compared to the total volume of the container.
2. There are no intermolecular attractive or repulsive forces between the gas particles.

Evaluate real gas deviations

Under non-ideal conditions (such as high pressure or low temperature), intermolecular attractions become significant, drawing molecules closer together and reducing the effective volume. This causes the real density to be higher than the calculated ideal density. Additionally, the finite volume of the molecules themselves reduces the free space, also leading to a higher actual density.
</reasoning>

<answer>

Question 1a

The density of Earth's atmosphere under these conditions is \(1.19\text{ g/L}\).

Question 1b

To assume ideal behavior, we assume that the gas particles have negligible volume and experience no intermolecular attractive or repulsive forces.

If the conditions were not ideal, intermolecular attractive forces would pull the molecules closer together, causing the actual volume to be smaller and the real density to be higher than the calculated ideal density.
</answer>

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Answer:

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<reasoning>

Identify given values and target variables

Using the Molar Mass of Gases and Ideal Gas Law Calculations knowledge points
\[

$$\begin{aligned} &M = 29.0\text{ g/mol}\\ &T = 298\text{ K}\\ &P = 1.00\text{ atm}\\ &R = 0.08206\text{ L}\cdot\text{atm}/(\text{mol}\cdot\text{K})\\ &\text{Target: Density } d = \frac{m}{V} \end{aligned}$$

\]

Derive the gas density formula

Using the Ideal Gas Law Calculations knowledge point
\[

$$\begin{aligned} &PV = nRT \implies PV = \frac{m}{M}RT\\ &d = \frac{m}{V} = \frac{PM}{RT} \end{aligned}$$

\]

Calculate the density of the atmosphere

Using the Ideal Gas Law Calculations knowledge point
\[

$$\begin{aligned} &d = \frac{(1.00\text{ atm})(29.0\text{ g/mol})}{(0.08206\text{ L}\cdot\text{atm}/(\text{mol}\cdot\text{K}))(298\text{ K})}\\ &d \approx 1.1859\text{ g/L} \approx 1.19\text{ g/L} \end{aligned}$$

\]

Analyze assumptions for ideal behavior

To assume ideal behavior, we assume:

1. The volume of individual gas particles is negligible compared to the total volume of the container.
2. There are no intermolecular attractive or repulsive forces between the gas particles.

Evaluate real gas deviations

Under non-ideal conditions (such as high pressure or low temperature), intermolecular attractions become significant, drawing molecules closer together and reducing the effective volume. This causes the real density to be higher than the calculated ideal density. Additionally, the finite volume of the molecules themselves reduces the free space, also leading to a higher actual density.
</reasoning>

<answer>

Question 1a

The density of Earth's atmosphere under these conditions is \(1.19\text{ g/L}\).

Question 1b

To assume ideal behavior, we assume that the gas particles have negligible volume and experience no intermolecular attractive or repulsive forces.

If the conditions were not ideal, intermolecular attractive forces would pull the molecules closer together, causing the actual volume to be smaller and the real density to be higher than the calculated ideal density.
</answer>

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