This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.
ScanSolve AI Answer
Let's derive the relationship between density and pressure using the ideal gas equation. Step 1: Start with the ideal gas equation. The ideal gas equation is given by: PV = nRT where: P = pressure V = volume n = number of moles R = ideal gas constant T = temperature Step 2: Express the number of moles (n) in terms of mass (m) and molar mass (M). The number of moles (n) of a gas is its mass (m) divided by its molar mass (M): n = (m)/(M) Step 3: Substitute the expression for n into the ideal gas equation. PV = ((m)/(M))RT Step 4: Rearrange the equation to isolate the density term ( = (m)/(V)). Divide both sides by V: P = ((m)/(V))(RT)/(M) Recognize that (m)/(V) is the density (): P = (RT)/(M) Step 5: Rearrange to show density as a function of pressure. = P (M)/(RT) Step 6: Identify constant terms at a given temperature. For a given gas (meaning M is constant) and at a given temperature (T is constant), the terms (M)/(RT) are all constant. Let's call this constant k: k = (M)/(RT) Therefore, the equation becomes: = Pk This shows that density () is directly proportional to pressure (P) when the temperature and molar mass are constant. Thus, at a given temperature, the density of a gas is proportional to the gas pressure P. Last free one today — make it count tomorrow, or type /upgrade for unlimited.