Macbook Air shuts down if it is not plugged into the power outlet through the adapter. What is happening?

I have restored to Mountain Lion before reading your post. Since downgrading to original OS, the mba did not shut down immediately when unplugged 3 out of 4 times that i had unplugged it from power. It stayed for about 2 to 4 minutes but it still shut down on its own. Tried reseting SMC a couple of times but, it would still have a hard time starting up without being plugged into the power adapter. It took long to come on with just the battery and when it did come on with the battery the fan would be running hard and after a couple of minutes it would shut down on its own. I would like to add, since I need the charger adapter connected almost at all times the battery has remained at 100% through this process. Here is also the battery info you wanted:

I have not tried the hardware test at this time. I am thinking it more software than hardware. I had no problem before. This just recently started. I believe the battery is still good. I have another comment from MadMacs0 that I might have installed an iWorm Backdoor/Trojan/Botnet malware.

Is this the possible cause? Is his comment reliable?

No intention to offend in anyway MadMacs0. I just want to be certain I take the right steps to resolve this. But as I mentioned prior to seeing OGELTHORPE's post and yours, I had already restored the mba to original os Mountain Lion. would this have resolve what you are mentioning about the worm & malware issues on my laptop?

Here is the latest EtreCheck on my mba since restoration to MOUNTAIN LION. I am also able to unplugged without the it shutting down immediately. It seems to last longer and longer on battery before shutting down on its own. The 1st time i unplugged after restore to mountain lion it took about 1 minute before it turned off. The second time about 3. The next time about 9 minutes. But it still has a hard time turning on with just the battery. The fan seems to run hard on battery and can barely turn on that way and practically does not turn on with just battery. I still need it connected to adapter to turn on. Anyways, here is the latest EtreCheck:

EtreCheck version: 2.0.11 (98)

Report generated November 7, 2014 4:28:20 AM CST

Hardware Information:ℹ️

MacBook Air (13-inch, Mid 2013) (Verified)

MacBook Air - model: MacBookAir6,2

1 1.3 GHz Intel Core i5 CPU: 2-core

4 GB RAM

BANK 0/DIMM0

2 GB DDR3 1600 MHz ok

BANK 1/DIMM0

2 GB DDR3 1600 MHz ok

Bluetooth: Good - Handoff/Airdrop2 supported

Wireless: en0: 802.11 a/b/g/n/ac

Video Information:ℹ️

Intel HD Graphics 5000 - VRAM: 1024 MB

Color LCD 1440 x 900

System Software:ℹ️

OS X 10.8.5 (12F45) - Uptime: 0:19:14

Disk Information:ℹ️

APPLE SSD SM0128F disk0 : (121.33 GB)

S.M.A.R.T. Status: Verified

disk0s1 (disk0s1) <not mounted> : 210 MB

Untitled (disk0s2) / [Startup]: 120.47 GB (108.82 GB free)

Recovery HD (disk0s3) <not mounted> [Recovery]: 650 MB

USB Information:ℹ️

Apple Internal Memory Card Reader

Apple Inc. BRCM20702 Hub

Apple Inc. Bluetooth USB Host Controller

Thunderbolt Information:ℹ️

Apple Inc. thunderbolt_bus

Gatekeeper:ℹ️

Mac App Store and identified developers

User Login Items:ℹ️

None

Internet Plug-ins:ℹ️

QuickTime Plugin: Version: 7.7.1

JavaAppletPlugin: Version: 14.6.0 - SDK 10.8 Check version

3rd Party Preference Panes:ℹ️

None

Time Machine:ℹ️

Time Machine not configured!

Top Processes by CPU:ℹ️

3% WindowServer

1% fontd

0% blued

Top Processes by Memory:ℹ️

318 MB WebProcess

120 MB Safari

43 MB mds

39 MB WindowServer

34 MB Finder

Virtual Memory Information:ℹ️

2.59 GB Free RAM

1.07 GB Active RAM

184 MB Inactive RAM

447 MB Wired RAM

285 MB Page-ins

0 B Page-outs

I hope the downgrade to Mountain Lion at least removed the worm and adware. Thanks MadMacs0 for the info. Hope to see your comments tomorrow.

I did try the AHT. It showed no problem or issue concerning my MBA "ADP000". I am not sure what i else i can try other than opening the computer to check the physical connections of the battery and eventually spending for a replacement battery.

I hope a true solution to this can be found. I see others with similar problems posted but no one really offering a true solution for this particular scenario.

Tried to reset SMC many times but it didn't change anything. The battery however is no longer being charged by the adapter even the computer is plugged in. The light remains orange. I restored to one of my earlier back-ups on time machine with Mavericks and the iWorm and malware returned. So, I decided to erase the hard drive and restore to original OS. I was able to restore to Yosemite. Here is the result of the latest estrecheck and battery info:

EtreCheck version: 2.0.11 (98)

Report generated November 7, 2014 at 7:30:08 PM CST

Hardware Information:ℹ️

MacBook Air (13-inch, Mid 2013) (Verified)

MacBook Air - model: MacBookAir6,2

1 1.3 GHz Intel Core i5 CPU: 2-core

4 GB RAM Not upgradeable

BANK 0/DIMM0

2 GB DDR3 1600 MHz ok

BANK 1/DIMM0

2 GB DDR3 1600 MHz ok

Bluetooth: Good - Handoff/Airdrop2 supported

Wireless: en0: 802.11 a/b/g/n/ac

Video Information:ℹ️

Intel HD Graphics 5000 -

Color LCD 1440 x 900

System Software:ℹ️

OS X 10.10 (14A389) - Uptime: 0:8:24

Disk Information:ℹ️

APPLE SSD SM0128F disk0 : (121.33 GB)

S.M.A.R.T. Status: Verified

EFI (disk0s1) <not mounted> : 210 MB

Recovery HD (disk0s3) <not mounted> [Recovery]: 650 MB

Untitled (disk1) / [Startup]: 120.10 GB (109.93 GB free)

Core Storage: disk0s2 120.47 GB Online

USB Information:ℹ️

Apple Internal Memory Card Reader

Apple Inc. BRCM20702 Hub

Apple Inc. Bluetooth USB Host Controller

Thunderbolt Information:ℹ️

Apple Inc. thunderbolt_bus

Apple Inc. Thunderbolt to Gigabit Ethernet Adapter

Gatekeeper:ℹ️

Mac App Store and identified developers

User Login Items:ℹ️

None

Internet Plug-ins:ℹ️

Default Browser: Version: 600 - SDK 10.10

QuickTime Plugin: Version: 7.7.3

3rd Party Preference Panes:ℹ️

None

Time Machine:ℹ️

Skip System Files: NO

Auto backup: YES

Volumes being backed up:

Untitled: Disk size: 120.10 GB Disk used: 10.17 GB

Destinations:

Bogie Back-Up [Network]

Total size: 0 B

Total number of backups: 12

Oldest backup: 2014-08-19 18:08:04 +0000

Last backup: 2014-11-06 20:25:00 +0000

Size of backup disk: Too small

Backup size 0 B < (Disk used 10.17 GB X 3)

Top Processes by CPU:ℹ️

9% backupd

5% WindowServer

5% diskimages-helper

2% mds_stores

1% mds

Top Processes by Memory:ℹ️

180 MB com.apple.WebKit.WebContent

76 MB Finder

69 MB mds_stores

52 MB backupd

52 MB Safari

Virtual Memory Information:ℹ️

1.83 GB Free RAM

1.38 GB Active RAM

444 MB Inactive RAM

644 MB Wired RAM

1.86 GB Page-ins

262 KB Page-outs

At this point, the computer works fine with the adapter plugged in, light remains orange, the battery is not being charged. I believe the computer will still turn off once I unplugged the adapter from the computer although the battery is supposedly 88% charged.

If this is a hardware problem I hope it's just the battery. I believe that's the next step I have to take. Unless anyone has any other ideas.

I no longer have warranty on this mba. I still have the problem of shutting down when unplugged from the adapter. Since no one has suggested anything else that I can do to try to resolve this, I'm only left with the option of replacing the battery. I would like to order the correct battery for my mba (Mid-2013 MBA 13"). Does anyone know a place I can go to or order online from that provides the replacement battery if the OEM is too expensive? Is Amazon a good place to order and which brand?

Everyone thanks for all the help. Especially to MacMacs0, who informed me of the iWorm & Adware on my mba. I was able to take the steps to eradicate them. I also learned that you don't have to wipe out your hold hard drive & restore to get rid of that iWorm. You can also simply delete "com.JavaW.plist" and the "JavaW" folder that goes with. I think that eliminates the iWorm altogether.

As for the problem that I was mainly concerned with "MBA shuts down when unplugged although battery is at 100% charge and only has 137 cycle life" this is also solved. Once, i opened the MBA, i noticed that the battery connector was not fully seated to the logic board. I simply had to lock it in place and the laptop works perfect with the battery. How it came loose, your guess is as good as mine.

I meant "whole hard drive"