Problem 34 - University Physics with Modern Physics

Question

The conditions of standard temperature and pressure (STP) are a temperature of \(0.00^{\circ} \mathrm{C}\) and a pressure of 1.00 \(\mathrm{atm}\) . (a) How many liters does 1.00 \(\mathrm{mol}\) of any ideal gas occupy at STP? (b) For a scientist on Venus, an absolute pressure of 1 Venusian-atmosphere is 92 Earth- atmospheres. Of course she would use the Venusian-atmosphere to define STP. Assuming she kept the same temperature, how many liters would 1 mole of ideal gas occupy on Venus?

Step-by-Step Solution

Answer

(a) 22.4 L at STP on Earth; (b) 0.244 L on Venus at 92 Earth-atmospheres.

Step 1: Calculate Volume at STP on Earth

At STP on Earth, 1 mole of an ideal gas occupies 22.4 liters. This is a standard result derived from the Ideal Gas Law, \(PV = nRT\), where \(P = 1.00\,\mathrm{atm}\), \(T = 273.15\,\mathrm{K}\), and \(R\) is the gas constant.

Step 2: Determine Pressure on Venus

On Venus, 1 Venusian-atmosphere is equivalent to 92 Earth-atmospheres. Thus, if the pressure is 1 Venusian-atmosphere, this translates to a pressure of \(92\,\mathrm{atm}\) in Earth terms.

Step 3: Apply Ideal Gas Law on Venus

Using the Ideal Gas Law again, \(PV = nRT\). Here, \(P = 92\,\mathrm{atm}\), \(n = 1\,\mathrm{mol}\), \(R = 0.0821\,\mathrm{L}\cdot\mathrm{atm}\cdot\mathrm{mol^{-1}}\cdot\mathrm{K^{-1}}\), and \(T = 273.15\,\mathrm{K}\) (since the temperature remains the same on Venus).Rearrange to find \(V\):\[ V = \frac{nRT}{P} = \frac{1\times 0.0821\times 273.15}{92} \approx 0.244\,\mathrm{L} \]

Step 4: Compare Volume on Earth and Venus

On Earth, 1 mole of gas at STP occupies 22.4 L, whereas on Venus with the same temperature but a higher pressure, it occupies approximately 0.244 L.

Key Concepts

standard temperature and pressure (STP)

Standard Temperature and Pressure (STP) is a set of conditions used in gas calculations to make it easier to compare results from different experiments. It is defined as a temperature of 0.00°C (273.15 K) and a pressure of 1.00 atm (atmosphere). Under these conditions, scientists and students can predict the behavior of gases using the Ideal Gas Law. This standardization allows for consistent and comparable results worldwide.

• Temperature: 0°C
• Pressure: 1 atm

Applying the Ideal Gas Law at STP simplifies the calculation of the volume of an ideal gas. For instance, 1 mole of an ideal gas is known to occupy 22.4 liters at STP.

volume of ideal gas

The volume of an ideal gas depends on the conditions under which the gas is measured. The Ideal Gas Law equation, \(PV = nRT\), allows us to calculate this volume by using the pressure (\(P\)), number of moles (\(n\)), the gas constant (\(R\)), and temperature (\(T\)).

• P: Pressure of the gas
• V: Volume of the gas
• n: Moles of the gas
• R: Gas constant (0.0821 L·atm·mol^-1·K^-1)
• T: Temperature in Kelvin

At STP, the volume occupied by 1 mole of any ideal gas is approximately 22.4 liters. This is not just an arbitrary number but a result derived from the Ideal Gas Law applied under these specific standard conditions.

pressure conversions

Pressure conversions are crucial in chemistry, especially when working with gases from different environments. Pressure is often measured in units of atmospheres (atm), but it can also be expressed in other units such as Pascals (Pa) or mmHg. Converting between these units helps scientists work with the pressure in terms that make the most sense for their particular application.

Converting Between Units

When conducting experiments on Venus, the pressure is considered as 1 Venusian-atmosphere, equivalent to 92 Earth-atmospheres. Therefore, understanding how to convert between these units is important.

• 1 Venusian-atmosphere = 92 Earth-atmospheres

This conversion allows for applying the same formulas, such as the Ideal Gas Law, accurately regardless of the given planetary conditions.

Venusian-atmosphere

The Venusian-atmosphere serves as a fascinating departure from Earth's atmospheric conditions. On Venus, the pressure is significantly higher; 1 Venusian-atmosphere is equal to 92 Earth-atmospheres. This means that the pressure experienced on the surface of Venus is much greater than what we experience on Earth.

Implications for Gas Volume

Because of this high pressure, using the Ideal Gas Law with Venusian conditions provides interesting results. For instance, when calculating the volume of a gas on Venus at STP, the same 1 mole of ideal gas that occupies 22.4 liters on Earth will occupy a much smaller volume, about 0.244 liters, due to the increased pressure.

• Pressure Effect: Higher pressure reduces gas volume

This is a clear demonstration of how changing one variable in the Ideal Gas Law affects the others, showcasing the adaptability of gases to different environmental conditions.