Calculate The New Volume If 12.78 L Of A Gas At -50*C Is Heated To A Temperature Of 28*C

Calculate The New Volume If 12.78 L Of A Gas At -50°C Is Heated To A Temperature Of 28°C

Understanding how gases behave when subjected to temperature changes is fundamental in chemistry and physics. When a gas is heated or cooled, its volume typically changes, provided the pressure remains constant. This relationship is described by Charles's Law, a key principle in understanding gas laws. In this article, we will explore how to calculate the new volume of a gas when its temperature changes from -50°C to 28°C, starting with an initial volume of 12.78 liters.

Understanding Gas Behavior and Charles's Law

Before delving into the calculation, it’s important to understand the core concept behind the problem—Charles's Law.

What Is Charles's Law?

Charles's Law states that, at constant pressure, the volume of a fixed amount of gas is directly proportional to its temperature in Kelvin. Mathematically, it is expressed as:

\[ \frac{V1}{T1} = \frac{V2}{T2} \]

Where:

• \(V_1\) = initial volume


• \(T_1\) = initial temperature in Kelvin


• \(V_2\) = final volume


• \(T_2\) = final temperature in Kelvin

This law implies that if you increase the temperature of a gas, its volume increases proportionally, assuming the pressure remains constant.

Converting Temperatures to Kelvin

Since Charles’s Law requires temperatures in Kelvin, the first step is converting the given Celsius temperatures:

• Initial temperature, \(T_1\): -50°C


• Final temperature, \(T_2\): 28°C

The conversion formula:

\[ T(K) = T(°C) + 273.15 \]

Applying this:

• \(T_1 = -50 + 273.15 = 223.15\,K\)


• \(T_2 = 28 + 273.15 = 301.15\,K\)

Calculating the New Volume

Using the formula derived from Charles’s Law:

\[ V2 = V1 \times \frac{T2}{T1} \]

Plugging in the known values:

\[ V_2 = 12.78\,L \times \frac{301.15\,K}{223.15\,K} \]

Calculating the ratio:

\[ \frac{301.15}{223.15} \approx 1.349 \]

Now, multiply to find the new volume:

\[ V_2 = 12.78\,L \times 1.349 \approx 17.23\,L \]

Therefore, the gas's volume increases to approximately 17.23 liters when heated from -50°C to 28°C.

Step-by-Step Summary of the Calculation

To clarify the process, here is a step-by-step outline:

1. Identify known values:

• Initial volume (\(V_1\)): 12.78 L


• Initial temperature (\(T_1\)): -50°C


• Final temperature (\(T_2\)): 28°C

1. Convert temperatures to Kelvin:

• \(T_1 = -50 + 273.15 = 223.15\,K\)


• \(T_2 = 28 + 273.15 = 301.15\,K\)

1. Apply Charles's Law:

• \(V2 = V1 \times \frac{T2}{T1}\)

1. Calculate the ratio:

• \(\frac{301.15}{223.15} \approx 1.349\)

1. Compute the new volume:

• \(V_2 = 12.78 \times 1.349 \approx 17.23\,L\)

Additional Considerations in Gas Volume Calculations

While the calculation above provides a straightforward method, there are additional factors and scenarios to consider:

Pressure Conditions

• The calculation assumes constant pressure. If pressure varies, Boyle’s Law or combined gas laws must be used.
• Real-world applications often involve pressure changes, requiring more complex calculations.

Ideal Gas Behavior

• The calculations assume gases behave ideally.
• Deviations can occur at high pressures or low temperatures, affecting accuracy.

Practical Applications

• This calculation is useful in engineering, meteorology, and laboratory settings.
• Understanding gas volume changes is vital for designing chemical reactors, breathing apparatus, and weather predictions.

Common Mistakes to Avoid

When performing such calculations, be cautious of:

• Forgetting to convert Celsius to Kelvin.


• Using the wrong ratio or misapplying the law.


• Assuming pressure changes when the problem specifies constant pressure.


• Rounding errors—use sufficient decimal places for accuracy.

Summary and Key Takeaways

• Gas volume changes proportionally with temperature when pressure remains constant.


• Convert all temperatures to Kelvin before calculation.


• Use the formula \(V2 = V1 \times \frac{T2}{T1}\) to find the new volume.


• For the given problem, the volume increases from 12.78 L at -50°C to approximately 17.23 L at 28°C.

Conclusion

Understanding how to calculate the change in gas volume with temperature is a foundational skill in chemistry and physics. By applying Charles's Law and carefully converting temperatures, you can accurately determine how gases expand or contract under different thermal conditions. This knowledge is essential for various scientific and industrial applications, ensuring safe and efficient designs and processes.

Remember: Always verify that pressure remains constant or adjust calculations accordingly, and consider real-world deviations from ideal behavior for precise measurements.