Problem 13 - University Physics with Modern Physics

Question

If a certain amount of ideal gas occupies a volume \(V\) at STP on earth, what would be its volume (in terms of \(V )\) on Venus, where the temperature is \(1003^{\circ} \mathrm{C}\) and the pressure is 92 atm?

Step-by-Step Solution

Answer

The gas volume on Venus would be approximately \(\frac{V}{4.675}\).

Step 1: Understand the Problem and Given Conditions

We need to find the volume of an ideal gas on Venus in terms of its original volume at Standard Temperature and Pressure (STP) on Earth. At STP, temperature is 273.15 K and pressure is 1 atm. On Venus, the temperature is 1003°C, which is equivalent to 1276.15 K, and the pressure is 92 atm.

Step 2: Use Ideal Gas Law to Relate Conditions

The Ideal Gas Law is given by \(PV = nRT\). Because the amount of gas and the gas constant \(R\) are constant, we can set up a ratio between the initial (Earth) and final (Venus) conditions: \(\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}\). We will solve for the unknown \(V_2\).

Step 3: Substitute Known Values into the Equation

Use the initial conditions \(P_1 = 1\) atm, \(V_1 = V\), \(T_1 = 273.15\) K, and final conditions \(P_2 = 92\) atm, \(T_2 = 1276.15\) K. Substitute these into the formula: \(\frac{1 \cdot V}{273.15} = \frac{92 \cdot V_2}{1276.15}\).

Step 4: Solve for the Final Volume \(V_2\)

Rearrange the equation to solve for \(V_2\): \(V_2 = \frac{V \cdot 273.15 \cdot 92}{1276.15}\). Calculate \(V_2\) using these values to express it in terms of \(V\): \(V_2 = \frac{V \cdot 92}{4.675}\), simplifying this gives \(V_2 \approx \frac{V}{4.675}\).

Key Concepts

Standard Temperature and Pressure (STP)

Understanding Standard Temperature and Pressure, or STP, is crucial for solving gas law problems. STP is a set of agreed-upon conditions used to make comparisons between different sets of measurements.
In chemistry, STP is defined as a temperature of 273.15 K (0°C) and a pressure of 1 atm. These conditions are used as a reference point for the behavior of gases, known as ideal gases.
The behavior of these gases under STP can be predicted using the Ideal Gas Law, which enables us to calculate volumes, pressures, or temperatures by knowing at least one variable.

Temperature and Pressure on Venus

The environmental conditions on Venus are drastically different from those on Earth.
The temperature on Venus can soar up to 1003°C, which is converted to Kelvin as 1276.15 K. This extremely high temperature is due to the thick, greenhouse gas-rich atmosphere trapping heat.
Furthermore, the pressure on Venus is about 92 atm, much higher than Earth's atmospheric pressure. Such extreme conditions affect the behavior of gases significantly.

• The high temperature increases the energy of gas molecules, causing them to move more vigorously.
• The high pressure compresses gases into much smaller volumes compared to more moderate conditions like those on Earth.

Volume Comparison of Gases

Comparing the volumes of gases under different conditions involves applying the Ideal Gas Law: \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is temperature.
When dealing with volume comparisons, we often use the ratio form of the Ideal Gas Law: \( \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \). This helps determine how a gas’s volume changes when moved from one environment to another.
For instance, when calculating the volume of a gas on Venus in terms of STP volume from Earth, the conditions on Venus (high pressure and temperature) must be plugged into the equation.

• At STP: \( P_1 = 1 \) atm, \( T_1 = 273.15 \) K.
• On Venus: \( P_2 = 92 \) atm, \( T_2 = 1276.15 \) K.

By simplifying and rearranging the equation, we find that the volume on Venus \( V_2 \) becomes approximately \( \frac{V}{4.675} \), indicating a significantly reduced volume due to the extreme Venusian conditions.