Problem 27 - Introductory Chemistry Atoms First

Question

A balloon of methane gas, \(\mathrm{CH}_{4}\), has a temperature of \(-2.0^{\circ} \mathrm{C}\) and contains \(2.35 \mathrm{~g}\) of the gas. What is the temperature of the gas in Kelvin? How many moles of the gas does the balloon contain?

Step-by-Step Solution

Answer

The temperature of the methane gas in Kelvin is 271.15 K, and the balloon contains approximately 0.146875 moles of methane gas.

Step 1: Convert temperature from Celsius to Kelvin

Given temperature in Celsius (T_C) = -2.0°C To convert it into Kelvin, we need to add 273.15 to the given temperature. Temperature in Kelvin (T_K) = T_C + 273.15 Let's calculate the Temperature in Kelvin.

Step 2: Calculation of temperature in Kelvin

T_K = -2.0 + 273.15 T_K = 271.15 K Now we have found the temperature of the methane gas in Kelvin, which is 271.15 K.

Step 3: Find the number of moles of methаne gas

To find the number of moles of methаne gas, we will use the formula: Number of moles (n) = mass (m) / molar mass (M) Given mass (m) of methаne gas = 2.35 g Molar mass (M) of methаne gas, CH4 = 12 (Carbon) + 4 x 1 (Hydrogen) = 12 + 4 = 16 g/mol Now, let's calculate the number of moles of methаne gas.

Step 4: Calculation of number of moles of methаne gas

n = m / M n = 2.35 g / 16 g/mol n = 0.146875 moles (rounded to 6 decimal places) Now we have found the number of moles of methаne gas, which is approximately 0.146875 moles. In conclusion, the temperature of the methane gas in Kelvin is 271.15 K, and the balloon contains approximately 0.146875 moles of methane gas.

Key Concepts

Temperature Conversion

Temperature conversion is a common requirement in chemistry to switch between different temperature scales like Celsius and Kelvin. In this exercise, we're converting -2.0°C to Kelvin, which is essential for many scientific calculations as Kelvin is the SI base unit.- To convert Celsius to Kelvin, simply add 273.15 to the Celsius temperature.- The formula is: \[ T_K = T_C + 273.15 \] - For our exercise, the conversion is: \[ T_K = -2.0 + 273.15 = 271.15 \, \text{K} \] This conversion helps align with the absolute temperature scale, which is important especially when dealing with gas laws and other scientific measurements.

Methane Gas

Methane gas, \( \mathrm{CH}_4 \), is a colorless, odorless gas. It's a simple alkane and the principal component of natural gas, making it significant in both industrial uses and as a heating fuel. Here are a few critical properties:- Methane consists of one carbon atom and four hydrogen atoms. - It's a potent greenhouse gas with a higher impact on global warming than carbon dioxide.Understanding methane's properties is crucial in predicting its behavior as a gas, which includes calculations related to its molar mass and temperature changes during its use.

Molar Mass Calculation

Calculating the molar mass of a compound like methane involves adding up the atomic masses of all the atoms in its formula.- For methane, \( \mathrm{CH}_4 \): - Carbon (C) has an atomic mass of about 12 g/mol. - Hydrogen (H) has an atomic mass of about 1 g/mol. - Methane has one carbon and four hydrogen atoms.Here's how the calculation works:- Molar mass of methane: \[ M = (1 \times 12\, \text{g/mol}) + (4 \times 1\, \text{g/mol}) = 16\, \text{g/mol} \] The molar mass forms the basis of calculating the number of moles from a given mass, crucial for stoichiometric calculations.

Kelvin Scale

The Kelvin scale is an absolute temperature scale important for scientific experiments. - It starts from absolute zero, the point where particles theoretically have no kinetic energy. - Unlike Celsius, Kelvin does not use degrees. The change of 1 Kelvin is equivalent to a change of 1°C. In molecular and atomic scale work, Kelvin ensures consistency and precision. This is important when working with gases as per Charles's Law or the Ideal Gas Law, which involve temperature-dependent calculations. The Kelvin temperature used in this problem is a key factor in deriving the number of moles of methane gas when related to gas equations.