Question

A container contains certain gas of mass \(m\) at high pressure. Some of the gas has been allowed to escape from the container. After some time, the pressure of the gas becomes half and its absolute temperature two-third. The amount of the gas escaped is (a) \(2 \mathrm{~m} / 3\) (b) \(\mathrm{m} / \mathrm{2}\) (c) \(\mathrm{m} / 4\) (d) \(m / 6\)

Short answer

None of the provided options are correct. Based on the calculation, the mass of the gas escaped is m/3, which is not listed.

Step-by-step solution

Identify Given Information and Relations

Start by noting down the information given. Here, we learn that the final pressure, P_f, is half the initial pressure, P_i. Similarly, the final absolute temperature, T_f, is two-thirds of the initial temperature, T_i. We also need to acknowledge the Ideal Gas Law for a fixed amount of gas, which states that P_iV_i/T_i = P_fV_f/T_f.

Applying the Ideal Gas Law to Initial and Final States

Since the pressure is halved and the temperature is reduced to two-thirds, we can relate the initial and final states by the formula P_i * V_i / T_i = P_f * V_f / T_f. Substituting the relational changes into the equation becomes 1 * V_i / 1 = 1/2 * V_f / (2/3), where V_i is the initial volume and V_f is the final volume of the gas.

Solve for the Final Volume V_f

Solve the relation by isolating Vf to understand the change in volume. (1 * V_i / 1) = (1/2 * V_f / (2/3)), which simplifies to V_i = (3/4) * V_f after cross multiplication and simplifying.

Calculate the Amount of Gas Escaped

Recognizing that the volume is directly proportional to the amount of gas by Avogadro's Law, we can say that m_i / m_f = V_i / V_f where m_i is the initial mass and m_f is the final mass. Since we know V_i = (3/4) * V_f, we can say that m_i / m_f = 3/4. Thus, m_f = (4/3) * m_i. The mass of gas escaped is m_i - m_f = m_i - (4/3) * m_i.

Final Calculation of the Mass of Gas Escaped

Compute the mass of the escaped gas m_escaped = m_i - (4/3) * m_i, which results in m_escaped = m_i - 4m_i / 3 = -1/3 * m_i. Taking the absolute value and considering that mass cannot be negative, the amount of the gas escaped is 1/3 * m_i. Therefore, the answer is (1/3)*m = m/3. However, this result is not an answer choice, indicating a possible error in the initial assumption or the answer choices.

Key concepts

Chemical Thermodynamics

Understanding chemical thermodynamics is critical when dealing with problems involving gas laws and reactions. Thermodynamics is a branch of physical science that deals with the relations between heat and other forms of energy. In the context of chemical reactions and processes, it specifically looks at the transfer of energy from one form to another and between substances.

In the exercise provided, thermodynamics helps us understand the changes in pressure (\( P \)) and absolute temperature (\( T \) as variables that affect the state of the gas within a system. According to the first law of thermodynamics, also known as the law of energy conservation, the energy of a closed system must be constant. When some of the gas escapes the container, the system does work on the surrounding environment. This causes changes in the internal energy of the gas, which in turn affects the pressure and temperature.

Further, when dealing with gases, we often consider the Ideal Gas Law, which is a good approximation under many conditions and facilitates understanding the relationship between pressure, volume, temperature, and number of moles of gas. For the given problem, using the Ideal Gas Law simplifies the complex inter-relationship of the variables and allows students to calculate the mass of the escaped gas upon changes in temperature and pressure.

Physical Chemistry

The exercise involves concepts from physical chemistry, a branch of chemistry that deals with the physical properties of molecules, the forces that act upon them, and their interactions. Specifically, it's the study of how matter behaves on a molecular and atomic level and how chemical reactions occur. Understanding the physical behavior of gases is a significant topic within physical chemistry, often explained through various gas laws.

Application in the Exercise

Applying physical chemistry principles, notably the Ideal Gas Law, to the exercise allows for the prediction of the behavior of a gas under different conditions. This law combines several separate gas laws like Boyle's Law, Charles's Law, and Avogadro's Law into one fundamental equation: \(PV = nRT\). Here, \(P\) denotes the pressure of the gas, \(V\) its volume, \(n\) the amount in moles, \(R\) the gas constant, and \(T\) the absolute temperature. By analyzing how changing one variable affects the others, we can understand the behavior of gas in the exercise—leading to conclusions about the change in mass due to pressure and temperature changes.

Avogadro's Law

In the context of the exercise, Avogadro's Law plays a crucial role in finding the solution to the problem. Avogadro's Law states that equal volumes of all gases, at the same temperature and pressure, contain the same number of molecules. In other words, the volume (\(V\)) of a gas is directly proportional to the number of moles (\(n\)) if the pressure and temperature remain constant: \(V \propto n\).

Significance in the Solution Process

When we use this law in combination with the Ideal Gas Law, it enables us to derive relationships between the volume and the amount (number of moles or mass) of gas before and after some of the gas has escaped. According to the law, if the volume of the gas decreases (as some gas escapes), the number of moles of gas must also decrease proportionally—assuming the pressure and temperature are held constant. However, in this situation pressure and temperature also change, which requires us to apply the Ideal Gas Law to fully characterize the state of the gas before and after. By understanding Avogadro's Law, students can better grasp why the volume and amount of the gas are related and how this affects the calculations involving gases under different conditions.