Question

A gas evolved during the fermentation of glucose (wine making) has a volume of \(0.67 \mathrm{~L}\) at \(22.5^{\circ} \mathrm{C}\) and \(1.00 \mathrm{~atm} .\) What was the volume of this gas at the fermentation temperature of \(36.5^{\circ} \mathrm{C}\) and 1.00 atm pressure?

Short answer

The gas volume at 36.5°C and 1.00 atm is approximately 0.701 L.

Step-by-step solution

Understand the Problem

The problem involves gas that is initially at a specific volume, temperature, and pressure (0.67 L, 22.5°C, and 1.00 atm). We need to find the new volume at a different temperature (36.5°C), while pressure remains unchanged.

Identify the Gas Law

Since the pressure remains constant, we can use Charles's Law, which is the relationship between volume and temperature: \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \). Given that the pressure is constant, this law will allow us to solve for the new volume, \( V_2 \).

Convert Temperatures to Kelvin

To use Charles's Law, temperatures must be in Kelvin. Convert each Celsius temperature to Kelvin by adding 273.15. Thus, \(T_1 = 22.5 + 273.15 = 295.65\) K and \(T_2 = 36.5 + 273.15 = 309.65\) K.

Use Charles's Law to Solve for the New Volume

Substitute the known values into the Charles's Law equation: \( \frac{0.67 \text{ L}}{295.65 \text{ K}} = \frac{V_2}{309.65 \text{ K}} \). Solve for \(V_2\) by cross-multiplying and dividing: \(V_2 = 0.67 \times \frac{309.65}{295.65} \).

Calculate the New Volume

Perform the calculation: \(V_2 = 0.67 \times \frac{309.65}{295.65} \approx 0.701 \) L. Therefore, the volume of the gas at the fermentation temperature is approximately 0.701 L.

Key concepts

Gas Laws

Gas laws are a set of essential equations that describe the behavior of gases under different conditions. These laws are crucial for understanding how gas volumes, pressures, and temperatures interrelate. One prominent gas law is Charles's Law, which specifically deals with the relationship between the volume and temperature of a gas when the pressure is held constant. According to Charles's Law, for a given amount of gas at constant pressure, the volume is directly proportional to its absolute temperature (measured in Kelvin).

The formula for Charles's Law is:

• \[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \]

This formula indicates that if the temperature of a gas increases, its volume will increase too, assuming the pressure remains unchanged. Understanding this concept is particularly useful for predicting how gases will behave in different scenarios, such as fermentation processes, air conditioning systems, and even hot air balloons. Recognizing the role of gas laws helps in making informed predictions and interpretations of phenomena involving gases.

Temperature Conversion

Temperature conversion is an important step in applying gas laws, especially since these laws require temperatures to be measured in Kelvin. The Kelvin scale is an absolute temperature scale, meaning it starts at absolute zero, which is the lowest possible temperature where all thermal motion ceases.

To convert a temperature from Celsius to Kelvin, you simply add 273.15 to the Celsius temperature. For example, converting 22.5°C to Kelvin involves adding 273.15, giving a result of 295.65 K. Similarly, for a temperature of 36.5°C, the Kelvin equivalent is 309.65 K.

The conversion is necessary because gas laws rely on absolute temperature to establish a direct relationship between volume and temperature. Since the Kelvin scale is linear and starts at zero, it ensures that temperature changes are accurately reflected in volume changes. This accuracy is crucial for scientific calculations, where precision is key.

Volume Calculation

Volume calculation in the context of gas laws involves using equations to determine how a gas's volume will change under different conditions. In the exercise, we see the use of Charles's Law, where the known initial volume and temperature, along with a new temperature, allow us to find the new volume.

The process involves:

• Identifying the initial conditions: \(V_1 = 0.67\) L and \(T_1 = 295.65\) K.
• Substituting these values, along with the new temperature \(T_2 = 309.65\) K, into the Charles's Law formula.
• Solve for the unknown volume \(V_2\), which represents the new volume under changed temperature conditions.

The calculation results in the new volume, which in this case is approximately 0.701 L. This method illustrates the importance of understanding how various factors affect gas behavior. Knowing how to calculate volume changes is not just academic; it can apply in practical situations such as chemical processing or meteorology, where predicting gas volume is essential.