Question - Ideal Gas Law and Pressure Change Relationship
Solution:
The image contains a question that references the ideal gas law in the form of PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvin. The question asks to show the change in pressure (∆P) in terms of the change in volume (∆V) and temperature.To solve this, let's start with the given:PV = nRTLet's say the initial conditions are P, V, and T, and after the change, the conditions are P + ∆P, V + ∆V, and T. It is given that T (temperature) is constant, so n, R, and T remain unchanged. According to the ideal gas law:P * V = (P + ∆P) * (V + ∆V)Since nRT is constant, we can set both sides of the equation equal to nRT.PV = nRT(P + ∆P)(V + ∆V) = nRTSince both equations are equal to nRT:PV = (P + ∆P)(V + ∆V)Expand the right side:PV = PV + P∆V + V∆P + ∆P∆VSince PV on both sides cancel out, we're left with:0 = P∆V + V∆P + ∆P∆VIf we assume that the changes in pressure and volume are small such that ∆P∆V is negligible compared to the other terms (which is a common approximation in physics), we can simplify the equation to:0 = P∆V + V∆PNow we can solve for ∆P in terms of ∆V:V∆P = -P∆V∆P = -(P/V) ∆VThis equation shows the change in pressure in terms of the initial pressure and volume and the change in volume, assuming constant temperature. This relation is consistent with Boyle's law, which states that for a given mass of an ideal gas at constant temperature, the product of pressure and volume is always constant.
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