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Physics 114 – Lecture 39

Physics 114 – Lecture 39. §13.6 The Gas Laws and Absolute Temperature Boyle’s Law : ( ~1650) − For a sample of gas, for which T = const, V 1/P or PV = const Effect of Temperature? Charles’ Law : ( ~1780) – For a sample of gas, for which P = constant, V T, if we

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Physics 114 – Lecture 39

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  1. Physics 114 – Lecture 39 • §13.6 The Gas Laws and Absolute Temperature • Boyle’s Law: (~1650) − For a sample of gas, for which T = const, V 1/P or • PV = const • Effect of Temperature? • Charles’ Law: (~1780) – For a sample of gas, • for which P = constant, V T, if we redefine the origin of T → T(K) = T(0C) + 273.15 – absolute or Kelvin scale L39-s1,8

  2. Physics 114 – Lecture 39 • Gay-Lussac’s Law: (~1820) −For a sample of gas, for which V = const, P T, where T is in kelvins (K) • Study Example 13.9 • §13.7 The Ideal Gas Law • PV T • Amount of gas? Expt.: if P and T are const, V m • PV mT • Constant? → PV = α RT, where α depends on m • It turns out that α is conveniently expressed in moles L39-s2,8

  3. Physics 114 – Lecture 39 • E.g., the number of moles in 96.0 g of O2 for which the molecular mass is 2 X 16.0 = 32.0 • The Ideal Gas Law then becomes, • PV = nRT where R = 8.314 J/(mol. K) where R is the universal gas const and is the same for all gases L39-s3,8

  4. Physics 114 – Lecture 39 • Reminder: P is the absolute pressure and T, the temperature, is measured in kelvins • Of course real gases, as opposed to ideal gases, follow this law only when they are neither at very high pressures nor near their liquefaction point • §13.8 Problem Solving with the Ideal Gas Law • Study Problems 13.10, 13.11, 13.12 and 13.13 L39-s4,8

  5. Physics 114 – Lecture 39 • §13.9 Ideal Gas Law in Terms of Molecules: Avogadro’s Number • Avogadro’s Hypothesis: Equal volumes of gas at the same temperature and pressure contain equal numbers of molecules • This is consistent with R being the same for all gases • Thus: PV = nRT states that, if P, V and T are the same for samples of two different gases, then n must be the same for these gases since R has the same value for all gases and the number of molecules in 1 mole is the same for all gases L39-s5,8

  6. Physics 114 – Lecture 39 • The number of molecules in one mole of any pure substance is given by Avogadro’s number, NA • The accepted value is: • NA = 6.02 X 1023 molecules/mole • We have PV = nRT = • which may be written, PV = N k T • where • and where k is known as the Boltzmann constant L39-s6,8

  7. Physics 114 – Lecture 39 • §13.10 Kinetic Theory and the Molecular Interpretation of Temperature • Assumptions: • 1. Large number of mols each of of mass, m, moving randomly • 2. Mols on average far apart wrt their diameter – force between mols = 0, unless they are colliding • 3. Mols interact only when they collide and follow laws of classical mechanics • 4. Collisions with the container walls are elastic and of short duration, compared with time between collisions L39-s7,8

  8. Physics 114 – Lecture 39 x vx • Consider one molecule colliding with the wall Δp1 = -mv1x – (mv1x) = -2 mv1x for mol Δt = 2l/v1x F1 = Δp1/Δt = -2mv1x/ (2l/v1x) = -mv1x2/l For N molecules, total force on wall F = (m/l) (v1x2 + v2x2 + v3x2 + … + vNx2) Since v1x2 + v2x2 + v3x2 + … + vNx2 = N (vx2)ave F = (m/l) N (vx2)ave With (vx2)ave = v2ave /3 → P = F/A = ⅓ Nm v2ave /(Al) With V = Al → PV = ⅓ Nm v2ave = ⅔N(½ mv2ave) = ⅔N KEave Comparing with PV = NkT → KEave = ½ mv2ave = (3/2) kT Thus T is a measure of KEave of the molecules in the sample -vx l L39-s8,8

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