Question

A new grill has a mass of \(30.0 \mathrm{kg}\). You put \(3.0 \mathrm{kg}\) of charcoal in the grill. You burn all the charcoal and the grill has a mass of \(30.0 \mathrm{kg} .\) What is the mass of the gases given off? Explain.

Short answer

The mass of the gases given off is \(3.0\,kg\). This is calculated using the law of conservation of mass, which states that the total mass of the system remains constant throughout the process. After finding the total initial mass of the grill and charcoal, we subtract the final mass of the grill to find the mass of the gases given off.

Step-by-step solution

Identify the given information

We are given the following: - Initial mass of the grill, \(m_g = 30.0\,kg\) - Mass of charcoal added, \(m_c = 3.0\,kg\) - Final mass of the grill, \(m_{fg} = 30.0\,kg\)

Calculate the total initial mass

We need to calculate the total initial mass of the system, which includes both the grill and the charcoal. We can do this by adding the mass of the grill and the mass of the charcoal together: Total initial mass, \(m_{total} = m_g + m_c\) Plugging in the given values, we get: \(m_{total} = 30.0\,kg + 3.0\,kg = 33.0\,kg\)

Use the conservation of mass

Since the mass is conserved, the total initial mass should be equal to the final mass: Total initial mass = Final mass of the grill + Mass of gases given off We can now write an equation to find the mass of the gases given off: Mass of gases given off, \(m_{gases} = m_{total} - m_{fg}\)

Calculate the mass of gases given off

Plugging in the values in the equation, we find the mass of the gases given off: \(m_{gases} = 33.0\,kg - 30.0\,kg = 3.0\,kg\) So, the mass of the gases given off is \(3.0\,kg\).

Key concepts

Mass Calculation

Mass calculation is an essential part of understanding physical and chemical processes. In our scenario, we begin with calculating the total mass of the system before any process occurs. This involves summing up the mass of the initial components. For example:

1. The grill has a mass of 30.0 kg.
2. We add charcoal to the grill, which has a mass of 3.0 kg.

The total initial mass of the setup, therefore, is obtained by combining these two values, resulting in \( m_{total} = 30.0\, \text{kg} + 3.0\, \text{kg} = 33.0\, \text{kg} \). These calculations help ensure we account for all components before any changes occur, such as burning the charcoal.

Chemical Reactions

Chemical reactions involve the rearrangement of atoms and molecules to form new substances. When charcoal burns in the presence of oxygen, a chemical reaction occurs. Burning involves a combustion process, where the carbon from the charcoal reacts with oxygen in the air, forming new substances like carbon dioxide and other gases.

This reaction can be represented by a simplified chemical equation: \( \text{C (charcoal)} + \text{O}_2 \rightarrow \text{CO}_2 + \text{other gases} \). Importantly, no atoms are lost during this process, which aligns with the principle of conservation of mass. While the physical state of the charcoal changes, the total mass remains the same when accounted for, including the gaseous products emitted.

Gas Production

Gas production during chemical reactions is an indication of mass conservation in action. Once the charcoal has burned completely, generating gases is evidence that matter has transformed, but has not disappeared. The total initial mass was 33.0 kg, and after the reaction, the grill weighs 30.0 kg. The missing 3.0 kg represents gases released into the atmosphere.

Gas production is crucial for verifying that the equation \( m_{total} = m_{fg} + m_{gases} \) holds true. Here, the mass of the gases given off is calculated by subtracting the mass of the grill post-reaction from the total initial mass. Thus, \( m_{gases} = 3.0\, \text{kg} \). This calculation confirms that despite visible changes, the mass is conserved across the chemical process.