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A 39.6-mL sample of gas is trapped in a syringe and heated from \(27^{\circ} \mathrm{C}\) to \(127^{\circ} \mathrm{C}\). What is the new volume (in \(\mathrm{mL}\) ) in the syringe if the pressure is constant?

Short Answer

Expert verified
The new volume is 52.77 mL.

Step by step solution

01

Understand the Problem

We're given a gas in a syringe initially at a certain temperature and volume. We need to find the new volume after the gas is heated, assuming the pressure remains constant.
02

Identify Relevant Formula

Under constant pressure, the relation between temperature and volume of a gas is described by Charles's Law: \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \), where \( V_1 \) and \( V_2 \) are initial and final volumes, and \( T_1 \) and \( T_2 \) are initial and final temperatures in Kelvin.
03

Convert Temperatures to Kelvin

Convert the temperatures from Celsius to Kelvin using \( T(K) = T(^\circ C) + 273.15 \). Thus, the initial temperature: \( T_1 = 27 + 273.15 = 300.15 \) K, and the final temperature: \( T_2 = 127 + 273.15 = 400.15 \) K.
04

Solve for New Volume

Using Charles's Law, plug in the known values: \( \frac{39.6}{300.15} = \frac{V_2}{400.15} \). Solving for \( V_2 \): \( V_2 = \frac{39.6 \times 400.15}{300.15} \).
05

Calculate the New Volume

Calculate \( V_2 \): \( V_2 = \frac{39.6 \times 400.15}{300.15} = 52.77 \) mL. Hence, the new volume of the gas at the higher temperature is 52.77 mL.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Temperature Conversion
When dealing with temperature in gas calculations, it's crucial to use the correct unit of measurement. Many students wonder why converting temperature from Celsius to Kelvin is necessary. The Kelvin scale is an absolute temperature scale starting at absolute zero, the lowest energy state possible. This aspect makes Kelvin an ideal choice for scientific calculations involving gases.

**Converting Celsius to Kelvin**
- To convert temperature from Celsius to Kelvin, simply add 273.15 to the Celsius temperature.
- For example, if the temperature is \(27^\circ C\), the conversion to Kelvin would be \(27 + 273.15 = 300.15\) K.
- Similarly, for \(127^\circ C\), it would convert to \(127 + 273.15 = 400.15\) K.

Remembering this quick conversion formula will ensure that you always have the correct format for your calculations. It's a simple yet essential step that sets the foundation for further analysis.
Gas Volume Calculation
Calculating gas volume changes requires understanding how gases respond to temperature changes under constant pressure. This principle is captured in Charles’s Law, which links temperature and volume.

**Understanding Charles's Law**
- Charles's Law states that \(\frac{V_1}{T_1} = \frac{V_2}{T_2}\), where \(V_1\) and \(V_2\) are the initial and final volumes, and \(T_1\) and \(T_2\) are the initial and final temperatures in Kelvin.
- When the temperature of a gas increases, so does its volume, provided that the pressure remains constant.

**Steps in Calculation**
- Convert all temperatures to Kelvin if not already in that unit.
- Rearrange Charles's Law to solve for the unknown volume: \(V_2 = \frac{V_1 \times T_2}{T_1}\).
- Substitute the known values and solve for \(V_2\).

By applying these straightforward steps, you can successfully determine how gas volume changes with temperature, ensuring accuracy in every calculation.
Ideal Gas Law
While Charles's Law is pivotal for understanding the relationship between temperature and volume, it's a component of a broader framework known as the Ideal Gas Law.
The Ideal Gas Law is represented as \(PV = nRT\). It incorporates pressure (\(P\)), volume (\(V\)), moles of gas (\(n\)), the ideal gas constant (\(R\)), and temperature (\(T\)). This formula brings together various aspects of gases, allowing for comprehensive calculations when more than two variables are involved.

**How It Relates to Charles's Law**
- Charles's Law is a specific case of the Ideal Gas Law where pressure and moles of gas remain constant.
- Therefore, it focuses on volume and temperature only, explaining why volume increases with temperature under constant pressure.

Understanding both Charles's Law and the Ideal Gas Law provides a solid foundation for more advanced studies in thermodynamics and chemistry, helping students solve a wide range of gas-related problems with confidence.

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Most popular questions from this chapter

Calculate the volume of methane, \(\mathrm{CH}_{4}\), measured at \(300 \mathrm{~K}\) and 825 torr, that can be produced by the bacterial breakdown of \(1.25 \mathrm{~kg}\) of a simple sugar. $$ \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6} \rightarrow 3 \mathrm{CH}_{4}+3 \mathrm{CO}_{2} $$

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