1. The document discusses methods for calculating the area of regions bounded by curves using integral calculus.
2. Six methods are presented for computing the area of regions bounded above and below by curves including the use of polar coordinates.
3. One example calculates the area between the curves y=x2 and y=2x from x=0 to x=2 as 8π/15 using the integral of the difference of the two curves.
1. The document discusses molecular physics, specifically the first law of thermodynamics regarding energy changes in processes. It defines internal energy (U) as the sum of kinetic energy from motion and potential energy from interactions between particles in a system.
2. It explains how internal energy depends on the state of the system and can change through work or heat transfer. The first law of thermodynamics states that energy change in a system equals the heat transferred plus the work done.
3. It also discusses how internal energy can change through work done by or on the system, and how this relates to pressure-volume work based on the gas laws.
1. The document discusses methods for calculating the area of regions bounded by curves using integral calculus.
2. Six methods are presented for computing the area of regions bounded above and below by curves including the use of polar coordinates.
3. One example calculates the area between the curves y=x2 and y=2x from x=0 to x=2 as 8π/15 using the integral of the difference of the two curves.
1. The document discusses molecular physics, specifically the first law of thermodynamics regarding energy changes in processes. It defines internal energy (U) as the sum of kinetic energy from motion and potential energy from interactions between particles in a system.
2. It explains how internal energy depends on the state of the system and can change through work or heat transfer. The first law of thermodynamics states that energy change in a system equals the heat transferred plus the work done.
3. It also discusses how internal energy can change through work done by or on the system, and how this relates to pressure-volume work based on the gas laws.
3. Àæèë [A]
Àëèâàà ìåõàíèê õºäºë㺺íèé ººð÷ëºëòººð
òîäîðõîéëîãäîíî.
Òåðìîäèíàìèê àæèë íü :
dA F dx= ⋅
F P S
dV S dx
dA F dx P S dx P dV
P
= ⋅
= ⋅
= ⋅ = ⋅ ⋅ = ⋅
− äàðàëò
⋅dA = P dV
0dV > õèé òýëýõ ¿åä dA >0 ãàäàãøàà àæèë õèéíý.(ýåðýã )
dV <0 õèé øàõàãäàõ ¿åä dA <0 ãàäíààñ àæèë õèéíý.(ñºðºã)
4. Äîòîîä ýíåðãè -
Àòîì ìîëåêóëóóäûí äóëààíû õºäºë㺺íä íººöëºãäºõ ýíåðãèéã
äîòîîä ýíåðãè ãýíý. Äóëààíû ýíåðãè íü àòîì ìîëåêóëóóäûí ÷ºëººíèé
çýðýãò æèãä õóâààðüëàãäàíà.
Íýã ìîëåêóëä íººöëºãäºõ ýíåðãè :
Äîòîîä ýíåðãèéí òîìú¸î:
äîòîîä ýíåðãè íü
èçîòåðì ïðîöåññò
õàäãàëàãäàíà áóñàä
ïðîöåññò ººð÷ëºãäºíº.
UΔ
2
i
U kT=
2
i
U N kT= ⋅
2 2 2
A
i i i
U N kT v N kT v RT
i
2
U v R T
= ⋅ = ⋅ = ⋅
äîòîîä ýíåðãèéí ººð÷ëº ëòíü:
Δ = ⋅ Δ
8. Òåðìîäèíàìèêèéí 1-ð õóóëèéã
èçîïðîöåññóóäàä õýðýãëýõ
T=const èçîòåðì
ïðîöåññ
2 2
2
1
1 1
1 1 2
1 1 1 1 1 1
10
0
2
ln( ) ln
V VA
V
V
V V
i
dU vR dT dT dU const
Q A
PV V
A dA PdV PV PV dV PV V PV
V V
δ δ
= ⋅ ⋅ → = → =
=
⎛ ⎞
= = → = → = = ⎜ ⎟
⎝ ⎠
∫ ∫ ∫
Òåðìîäèíàìèêèéí 1- ð õóóëü áè ÷ âýë :
2 2 2
1 1
1 1 1
2
1
ln ln ln
ln
V V P
A PV v RT v RT
V V P
P
Q A v RT
P
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
= = ⋅ = ⋅⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠ ⎝ ⎠
⎛ ⎞
= = ⋅ ⎜ ⎟
⎝ ⎠
9. Èçîõîð V=const
Ýíý ¿åä ýçýëõ¿¿í ººð÷ëºãäºõã¿é.
0dA PdV dV const A= → = → =
( )
2 2
1 1
2 1
0
2 2 2
T TQ
T T
Q dU
i i i
Q dQ dU vR dT vR T T vR T
δ =
= = = ⋅ = − = Δ∫ ∫ ∫
Òåðìîäèíàìèêèéí 1- õóóëèéã áè ÷ âýë :
10. Äóëààí áàãòààìæ
Áèåèéí òåìïåðàòóðûã íýãæýýð íýìýãä¿¿ëýõýä øààðäàãäàõ äóëààíû
õýìæýýã äóëààí áàãòààìæ ãýíý.
Íýãæ ìàññòàé áèåèéí õóâèéí äóëààí áàãòààìæ :
Ìîëèéí äóëààí áàãòààìæ:
Q
C
T
Δ
=
Δ
0
Q
C
T m
Δ
=
Δ ⋅
0M
Q C
C C M
T v v
M
Δ
= ⋅ = =
Δ ⋅
− ìîëèéí ìàññ
v - áîäèñûí õýìæýý
11. V=const ¿åèéí
ìîëèéí äóëààí áàãòààìæ
2 2
MV V
Q dU i vR i
C C R
T v dT v v
Δ
= = = = =
Δ ⋅ ⋅
2
V
i
Q U vR T vC T= Δ = Δ = Δ
12. Èçîáàp P=const
Äàðàëò ººð÷ëºãäºõã¿é ó÷ðààñ àæèë õèéãäýíý, äîòîîä ýíåðãè íü
ººð÷ëºãäîæ, äóëààí ººð÷ëºãäºíº. Òåðìîäèíàìèêèéí 1- ð õóóëü íü:
Ñèñòåìèéí àâñàí äóëààí íü äîòîîä ýíåðãèéí ººð÷ëºëò áîëîí ãàäàãøàà
õèéõ àæèëòàé òýíö¿¿.
Q dU Aδ δ= +
2
1
2
1
2 1
2 1
0
( )
( )
2 2 2
V
V
TU
V
T
A PdV P V V
i i i
U dU vR dT v R T T v R T vC T
= = −
= = ⋅ = − = Δ = Δ
∫
∫ ∫
13. Äàðàëò òîãìòîë ¿åèéí
Òåðìîäèíàìèêèéí 1- ð õóóëü
PV vRT
δ
=
Δ Δ
òýãøèòãýëèéã äèôôåðåíöèàëáàë :
VdP +PdV =vRdT
A = PdV =vRdT
A = P V =vR T
1 1
2 2 2
i i i
Q A U vR T vR T vR T A
⎛ ⎞ ⎛ ⎞
Δ = + Δ = Δ + Δ = + Δ = +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
14. P= const ¿åèéí
ìîëèéí äóëààí áàãòààìæ
2
2 2
2
2
2
MP P V
P P
V
Q i i
C C R R C R R
T v
i
Q vC T v T
CQ i
R
Δ +
= = = + = + =
Δ ⋅
+
= Δ = Δ
Δ
= =Èçîáàð ¿åä
A
P VC C R− =
Ìàéåðûí òýãøèòãýë
