🔑:## Step 1: Recall the Bohr model equation for the radius of the nth orbitThe radius of the nth orbit in the Bohr model is given by r_n = frac{n^2 hbar^2}{m k e^2}, where hbar is the reduced Planck constant, m is the mass of the electron, k is Coulomb's constant, e is the charge of the electron, and n is the principal quantum number.## Step 2: Recall the equation for the speed of the electron in the nth orbitThe speed of the electron in the nth orbit can be derived from the Bohr model by using the equation for the centripetal force, which is provided by the Coulomb force. This leads to the equation v_n = frac{k e^2}{n hbar}.## Step 3: Express the speed of the electron in terms of the fine structure constantThe fine structure constant alpha is defined as alpha = frac{k e^2}{hbar c}, where c is the speed of light. Rearranging this equation to solve for k e^2 gives k e^2 = alpha hbar c.## Step 4: Substitute the expression for k e^2 into the equation for v_nSubstituting k e^2 = alpha hbar c into the equation for v_n gives v_n = frac{alpha hbar c}{n hbar} = frac{alpha c}{n}.## Step 5: Consider the speed of the electron in a hydrogen-like atom with a nuclear charge of ZeFor a hydrogen-like atom with a nuclear charge of Ze, the Coulomb force is F = frac{k (Ze) e}{r^2}. Following similar steps as before, we find that the speed of the electron in the nth orbit is v_n = frac{k (Ze) e}{n hbar}. Substituting k e^2 = alpha hbar c and Z into this equation gives v_n = frac{Z alpha hbar c}{n hbar} = frac{Z alpha c}{n}.The final answer is: boxed{frac{Z alpha c}{n}}

🔑:The movement of particles in a gas and the temperature of the gas are closely related, and this relationship has a significant impact on the pressure exerted by the gas. Here's a detailed explanation:Kinetic Molecular Theory (KMT): According to the KMT, a gas consists of tiny particles (atoms or molecules) that are in constant random motion. The particles collide with each other and the walls of their container, exerting a force that we experience as pressure.Temperature and Particle Motion: The temperature of a gas is a measure of the average kinetic energy of its particles. As the temperature increases, the particles gain kinetic energy and move faster. Conversely, as the temperature decreases, the particles lose kinetic energy and move slower.Relationship between Temperature and Particle Motion: The relationship between temperature and particle motion can be summarized as follows:* Higher temperature: Particles move faster and have more kinetic energy.* Lower temperature: Particles move slower and have less kinetic energy.Effect on Pressure: The pressure exerted by a gas is directly related to the movement of its particles. As the particles collide with the walls of their container, they exert a force that contributes to the overall pressure. The pressure exerted by a gas is proportional to the frequency and force of these collisions.* Increased temperature: Faster-moving particles collide with the container walls more frequently and with greater force, resulting in an increase in pressure.* Decreased temperature: Slower-moving particles collide with the container walls less frequently and with less force, resulting in a decrease in pressure.Ideal Gas Law: The ideal gas law, PV = nRT, describes the relationship between the pressure (P), volume (V), and temperature (T) of a gas. The law states that, at constant volume, the pressure of a gas is directly proportional to its temperature.In summary, the movement of particles in a gas and the temperature of the gas are closely related. As the temperature increases, the particles move faster, resulting in more frequent and forceful collisions with the container walls, which increases the pressure exerted by the gas. Conversely, as the temperature decreases, the particles move slower, resulting in less frequent and less forceful collisions, which decreases the pressure exerted by the gas.

🔑:To solve this problem, we'll follow the steps below:## Step 1: Understanding the relationship between Kc and KpThe relationship between Kc (equilibrium constant in terms of concentration) and Kp (equilibrium constant in terms of partial pressures) is given by the equation Kp = Kc * (RT)^Δn, where R is the gas constant, T is the temperature in Kelvin, and Δn is the difference in the number of moles of gas between the products and reactants.## Step 2: Calculate Δn for the reactionFor the reaction N2(g) + 3H2(g) ⇌ 2NH3(g), the number of moles of gas on the reactant side is 1 (N2) + 3 (H2) = 4 moles, and on the product side is 2 (NH3) = 2 moles. Therefore, Δn = 2 - 4 = -2.## Step 3: Calculate Kp at 600 K from given KcFirst, we need to calculate Kp at 600 K using the given Kc (0.80) and the formula Kp = Kc * (RT)^Δn. The gas constant R is approximately 0.0821 L*atm/(mol*K). Therefore, Kp at 600 K = 0.80 * (0.0821 * 600)^(-2).## Step 4: Perform the calculation for Kp at 600 KKp at 600 K = 0.80 * (0.0821 * 600)^(-2) = 0.80 * (49.26)^(-2) = 0.80 * (0.000406) = 0.32448, but this step is to illustrate the process. The actual calculation should directly apply the formula correctly: Kp = 0.80 * (0.0821*600)^(-2) = 0.80 * (49.26)^(-2).## Step 5: Calculate the actual value for Kp at 600 K correctlyKp at 600 K = 0.80 * (49.26)^(-2) = 0.80 / (49.26)^2 = 0.80 / 2427.4976 = 0.0003297, approximately.## Step 6: Understanding the effect of temperature on KWhen the temperature increases, the value of Kp changes according to the formula Kp2 / Kp1 = exp(ΔH/R * (1/T1 - 1/T2)), where ΔH is the enthalpy change of the reaction, R is the gas constant, and T1 and T2 are the initial and final temperatures, respectively.## Step 7: Recognizing the direct relationship is not necessary for calculating the change in KSince we already have Kp values at two different temperatures (given Kp at 650 K is 0.15, and we can calculate Kp at 600 K), we can directly compare these values to understand the effect of temperature on the equilibrium constant without needing to calculate ΔH.## Step 8: Comparing Kp values to determine the effect on the reaction directionGiven Kp at 650 K = 0.15 and our calculated Kp at 600 K, we compare these values. However, the actual task is to understand how the change in temperature affects the reaction direction based on the change in Kp.## Step 9: Determining the direction of the reaction based on Kp changeA decrease in Kp (from 600 K to 650 K) indicates that the reaction shifts towards the reactants as temperature increases, meaning the reaction is exothermic.The final answer is: boxed{0.15}

🔑:## Step 1: Calculate the volume of the vessel in cubic feet to ensure consistency in units.First, convert the volume from cubic inches to cubic feet, since 1 cubic foot = 1728 cubic inches. Volume = 400 in^3 / 1728 in^3/ft^3 = approximately 0.2315 ft^3.## Step 2: Determine the heat transfer coefficient for the immersion heater.The heat transfer coefficient (h) depends on the flow regime (laminar or turbulent) and the properties of the fluid (air in this case). For a closed vessel with an immersion heater, we can assume natural convection. The heat transfer coefficient for natural convection in air can range from 0.5 to 10 W/m^2K, depending on the temperature difference and the geometry of the system. For simplicity, let's assume an average value of 5 W/m^2K.## Step 3: Calculate the surface area of the immersion heater.The surface area (A) of the immersion heater in contact with the air is crucial for heat transfer. Without specific dimensions for the heater, we'll assume it's a cylindrical shape with a height (H) equal to the diameter of the vessel (4 inches or 0.3333 feet) to maximize heat transfer. The radius (r) of the heater will be smaller than the vessel's radius to fit inside; assume it's half the vessel's radius, so r = 1 inch or 0.0833 feet. The surface area A = 2 * π * r * H = 2 * π * 0.0833 * 0.3333 ≈ 0.1759 ft^2.## Step 4: Calculate the power required to heat the air to the desired temperature.The power (P) required can be estimated using the formula P = m * c_p * ΔT / t, where m is the mass of air, c_p is the specific heat capacity of air (approximately 0.24 Btu/lb°F), ΔT is the temperature change, and t is the time. However, since we're given a 3500W heater, we'll focus on whether this power is sufficient to achieve the desired temperature change within a reasonable time frame.## Step 5: Estimate the mass of air in the vessel.The mass (m) of air can be calculated using the ideal gas law, PV = mRT, where P is pressure, V is volume, R is the gas constant, and T is temperature. At standard conditions (1 atm, 70°F), the density of air is approximately 0.0765 lb/ft^3. Thus, the mass of air in the vessel at standard conditions is m = density * volume = 0.0765 lb/ft^3 * 0.2315 ft^3 ≈ 0.0177 lb.## Step 6: Calculate the energy required to heat the air to 600°F.The initial temperature is assumed to be 70°F (standard room temperature). The temperature change ΔT = 600°F - 70°F = 530°F. The energy (Q) required is Q = m * c_p * ΔT = 0.0177 lb * 0.24 Btu/lb°F * 530°F ≈ 2.24 Btu.## Step 7: Convert the energy required into watt-hours to compare with the heater's power.1 Btu ≈ 1055 J, and 1 watt-hour (Wh) = 3600 J. Thus, the energy in watt-hours is 2.24 Btu * 1055 J/Btu / 3600 J/Wh ≈ 0.661 Wh.## Step 8: Calculate the time required to heat the air using the 3500W immersion heater.The time (t) required is t = energy / power. First, convert the power to watt-hours per second: 3500 W = 3500 Wh/s. Then, t = 0.661 Wh / 3500 Wh/s ≈ 0.000189 hours or approximately 0.68 seconds.## Step 9: Consider the heat transfer coefficient, insulation, and temperature control for uniform temperature distribution.To ensure a uniform temperature distribution, it's crucial to have good insulation to minimize heat loss and a sufficient heat transfer coefficient. The calculated heat transfer coefficient and the use of an immersion heater should facilitate relatively uniform heating. However, in practice, achieving perfect uniformity might require additional measures like stirring the air or using multiple heaters.## Step 10: Address the key factors affecting the final temperature.Key factors include the power of the heater, the heat transfer coefficient, insulation quality, and the initial conditions of the air. Given the high power of the heater relative to the small volume and mass of air, and assuming adequate insulation, the air should quickly reach the desired temperature. However, precise control of the final temperature might require a thermostat or temperature control system to prevent overheating.The final answer is: boxed{0.68}